Ballistic Transport of AlAs Two-Dimensional Electrons

Ballistic Transport of AlAs Two-Dimensional Electrons Oki Gunawan

Ballistic Transport of AlAs Two-Dimensional Electrons Oki Gunawan

A dissertation presented to the faculty of Princeton University in candidacy for the degree of Doctor of Philosophy Recommended for acceptance by the Department of Electrical Engineering

September 2007

© Copyright by Oki Gunawan, 2007. All rights reserved.

ABSTRACT The success of our modern electronics age stems from our advanced technology to process information by manipulating electrons in solid-state devices. Among the few fundamental ways to manipulate the electrons, such as using their charge and spin, the control and manipulation of the electron’s valley degree of freedom in semiconductors remains practically unexplored. In this thesis we focus on some basic aspects of ballistic transport in a two-valley two-dimensional electron system (2DES), realized in high quality AlAs quantum wells. We start by demonstrating valley-resolved ballistic transport in an experiment using a Hall bar device with a surface grating. From the analysis of the frequencies of the commensurability oscillations in the magnetoresistance at various densities we deduce the mass anisotropy factor, namely the ratio of the longitudinal and the transverse effective masses, ml / mt = 5.2 ± 0.5 , a fundamental parameter for the anisotropic conduction bands in AlAs. We then present results from similar experiments in devices with antidot lattices that reveal peaks in magnetoresistance. Through an analysis of the positions of the peaks associated with the smallest commensurate orbit, we obtain a value for the mass anisotropy factor, ml / mt = 5.2 ± 0.4 , consistent with the value deduced from the surface-grating samples. The anisotropy of the effective mass can be exploited to realize a simple "valley filter" device using a quantum point contact (QPC), a one-dimensional quantum ballistic channel. This device may play an important role in "valleytronics" or valley-based electronic applications. Our experiments on the QPC in the AlAs 2DES reveal that the conductance of this system is nearly quantized at multiples of 2e 2 / h , instead of 4e 2 / h as expected from a valley and spin degenerate system. This observation indicates a broken valley degeneracy due to the mass anisotropy as well as residual strain in the QPC.

Finally, we demonstrate a novel giant piezoresistance effect in an AlAs 2DES with an antidot lattice. Such a device may have potential applications as an ultra-sensitive strain sensor. It exemplifies one of the many uses of manipulating the electron valley degree of freedom in a solid-state device. iii

ACKNOWLEDGEMENTS The graduate school is such an arduous journey that has become possible because of the following people. I am grateful to have Prof. Mansour Shayegan as my advisor, his remarkable guidance and mentorship during my tenure in Princeton have made my graduate school years an extraordinary life experience. It is a rare and true privilege to be his student. I am thankful for Prof. Claire Gmachl and Prof. Stephen Lyon for their time and attention in reading my thesis. I would like to acknowledge Army Research Office and National Science Foundation for their generous support to our research. Shayegan’s group (mashgroup) comes with an interesting mix of people that surely have made my journey more enjoyable. The seniors: Tony Yau, Etienne, and Emanuel with whom I learnt the ropes and many valuable skills and styles to survive in the group. My contemporaries: Yakov, my lifestyle guru, a multi-talented person with a knack to code various killer applications in Matlab, notably the MASHMEASURE. It is a real privilege to have spent countless quality times with him discussing everything under the sun and to absorb some of his Matlab expertise. Kamran, my spiritual guru, with whom I learnt to develop critical eyes into experimental problems and data at hand and whose publication record subliminally provides spiritual guidance to the rest of us. Babur, I am thankful for his friendship and various help and for setting a new standard of grad student dress-code thus entitling him my fashion guru; Nathan, for his contagious enthusiasm and his expert guidance on American idioms, late night rides, warm family dinner and Grand Theft Auto sessions at his place; Shashank for his insightful advice on various matters, from circuit design to job hunting; and the younger members: Tayfun, Medini and Javad, I am thankful for their friendship and assistance to me on various occasions. I should also mention Eric Shaner of Lyon’s group who had given tremendous help in my early years. I thank the Graduate College, and for the people who run it, for many memorable moments that I spent in my first two years at Princeton: the awesome Gothic architecture, the Sunday brunch, the mesmerizing winter, the Friday social, the spellbinding Procter Hall’s organ tune on Sunday noon, and the wait for Vina. It was like living in a dream. I should mention the following people, Maw Lin Foo, (a.k.a. hpy / “the good friend of mine”), with whom I had shared many quality times in Graduate College talking about all sort of things and who had provided a steady dosage of Singaporean culture at Princeton. iv

His friendship has made my Princeton experience more enjoyable. It is a pleasure to have known these people: Wang Chih Chun, Yuan Yu, Jian Zhang, I-Chun, Guillaume Sabouret, and the Tsui’s group members: Gabor Csathy, Ravi, Amlan, Keji Lai, and Wanli Li, I am thankful for their friendship and help on various matters, Edith and Brian for their companionship in badminton courts, also my fellow graduate students in far away places: Hendra Kwee and Wirawan Purwanto at William and Mary, Wahyu Setyawan at Duke, Hery Susanto at NTU and then Lehigh, and Rizal Hariadi at Caltech. For my friends in greater Princeton area whom I and Vina have come to know, we are grateful for the times we spent together on various occasions and for the help that we received in many ways: Iksan and Fiona, Lea and Daniel, Melany and Ngiap Kie, Yenty, Rudy and Cisca, Olivia and Trogan, Esther and Robert, Wesin and Wenny, and not to forget, Hari Intan of Philadelphia for his enjoyable companionship on many weekend trips. For my Mom, thanks for everything and for the great upbringing she gave and my Dad for his support. My brothers and sister: Toto, Dede, Een, and Marlene, for the care and wonderful times I had at home. I would like to mention my friends in my teenage years: Johan Bulet, Melvyn, Ujuan, and Barnald in Jakarta; Way Kong, Tony Kuan and James Tan in Singapore that surely had made my life very colorful. Special thanks for Bryan Hoo and Tee Jong of the NTUCF who had made my journey to Princeton possible. Looking back, I have to mention my high school physics teacher, Drs. Zaharah Ramli, whose remarkable passion and enthusiasm had ignited a spark, the zeal for physics that continues to this day. As such is the influence of amazing teacher, a rare jewel and I was simply a very fortunate person that crossed her path. It is an honor to rest my acknowledgment to her in a place where Einstein and Feynman once walked and talked. I was again very fortunate to have met Prof. Yohanes Surya and Dr. Agus Ananda, who with selfless dedication and tons of energy had devoted much of their time, when they were graduate students, to teach me advanced level physics, a timeless rock-solid foundation. They are practically the ones responsible in charting my future trajectory – where here fourteen years later – I find myself defending my PhD thesis at Princeton. For Prof. Ooi Boon Siew that had shaped my early interest in semiconductor physics and provides valuable mentorship from time to time and also for Dr. Jurianto Joe that has been an exemplar and had helped fuel my determination to go for graduate school in US.

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Mostly I am grateful for Vina, for her enduring companionship and love, her homecooked meals, many memorable moments we had during my graduate school years, and for many trips together: the New Hope, the Six Flags, the Manhattan, and the APS trips. More importantly for her total dedication in taking the most time in rearing baby Nael in my final years at Princeton, also for my Mom-in-law for her generous help in Nael’s early years. Looking back, I am filled with gratitude and finally would like to thank God for His tremendous blessing and guidance to me in completing this journey.

vi

For Vina, Nael, and my teachers.

vii

CONTENTS ABSTRACT

iii

ACKNOWLEDGEMENTS

iv

CONTENTS

viii

LIST OF FIGURES

x

1. INTRODUCTION

1

2. BACKGROUND 2.1 Multivalley Semiconductors 2.2 AlAs Two-Dimensional Electron System 2.3 Ballistic Transport

5 7 11

3. EXPERIMENTAL DETAILS 3.1 Sample Fabrication 3.2 Device Measurement

13 19

4. COMMENSURABILITY OSCILLATIONS IN ALAS 2DES 4.1 4.2 4.3 4.4 4.5

Introduction Device Fabrication Experimental Results Analyses and Discussions Summary

22 25 26 28 33

5. ALAS 2D ELECTRONS IN AN ANTIDOT LATTICE 5.1 5.2 5.3 5.4 5.5 5.6

Introduction Device Fabrication Experimental Results Numerical Simulation Data Analysis Summary

34 36 37 39 42 46

6. QUANTUM POINT CONTACT IN ALAS 2DES 6.1 6.2 6.3 6.4

Quantum Point Contact Quantized Conductance Device Structure Analysis and Discussion

47 49 51 52

viii

6.5 Source Drain Bias Spectroscopy 6.6 Summary

60 63

7. ANOMALOUS GIANT PIEZORESISTANCE IN ALAS 2DES WITH ANTIDOT LATTICE 7.1 7.2 7.3 7.4 7.5 7.6

Introduction Device Fabrication Experimental Results and Discussions Discussions Device Characterization for Strain Sensor Application Summary and Conclusion

64 68 68 85 86 91

8. SUMMARY AND FUTURE PROJECTS 8.1 Summary of the Results 8.2 Possible Future Projects 8.3 Conclusion

93 95 97

APP. A LIST OF PUBLICATIONS

98

APP. B HALL BAR LITHOGRAPHY MASK FOR ALAS 2DES SAMPLES

100

APP. C SUMMARY OF ALAS EFFECTIVE MASS DETERMINATIONS

104

APP. D PIEZO STRAIN FACTOR CALIBRATION D.1 Piezo-actuator D.2 Strain Gauge D.3 Experimental Setup D.4 Piezo Strain Factor Characterization

108 110 112 116

APP. E CIRCUIT DIAGRAMS OF CUSTOM INSTRUMENTATIONS E.1 Bipolar Tunable Voltage Source E.2 Active Voltage Adder E.3 Programmable High Current Source

122 123 124

APP. F. A MODEL FOR THE ANOMALOUS GIANT PIEZORESISTANCE EFFECT IN ANTIDOT LATTICE

126

BIBLIOGRAPHY

132

ix

LIST OF FIGURES 1.1

Research perspective on the manipulation of electrons in solid-state devices based on the electrons’ properties such as charge, spin and valley degree of freedom. _________________________________________________ 2

2.1

Band structure of AlAs and the constant energy surface of a valley. ___________ 6

2.2

Constant energy surfaces in k-space for the conduction band edges of AlAs, Si and Ge. ___________________________________________________ 6

2.3

The layer structure of AlAs MODFET wafers used in this thesis: M415 (15 nm wide QW) and M409 (11 nm wide QW). _________________________ 8

2.4

Valley occupation in AlAs quantum wells and the 2D Fermi contours for narrow and wide wells. ______________________________________________ 8

2.5

Experimental setup for tuning the valley population in AlAs 2DES using a piezo-actuator. __________________________________________________ 10

2.6

Electron trajectories in diffusive and ballistic transport regimes. _____________ 12

3.1

The main steps of the AlAs device fabrication process. ____________________ 13

3.2

Schematic diagram of the Astex ECR-RIE system and the etching test pattern. __________________________________________________________ 17

3.3

A sample mounted on a piezo-actuator and the strain gauge to measure the applied strain (sample M409K8). __________________________________ 19

3.4

A typical experimental setup for device characterization showing the three main components._____________________________________________ 20

4.1

The original Weiss oscillations data and the conceptual description of the commensurability oscillations. _______________________________________ 23

4.2

A micrograph of a surface grating device, its device schematic and the diagrams for the X and Y valleys in k-space with their corresponding first two resonant orbits in real space. _____________________________________ 24

4.3

Commensurability oscillations and Shubnikov-de Haas data from M409N3 (a = 400 nm) and their corresponding Fourier spectra. _____________ 26

4.4

Commensurability oscillations at various densities (sample M409N3). ________ 27

4.5

Density dependence of the frequencies of the commensurability oscillations for X and Y valleys (sample M409N3). ______________________ 29

x

4.6

Commensurability oscillations data (a = 300 nm) and the corresponding Fourier spectrum (sample M415L3). __________________________________ 30

4.7

Inverse Fourier decomposition of the commensurability oscillations of Fig. 4.3 for the X and Y valleys. _____________________________________ 32

5.1

The data of the original antidot lattice experiment by D. Weiss. _____________ 35

5.2

The antidot lattice experiment in AlAs 2DES showing a micrograph of the antidot lattice region, the Fermi contours of the X and Y valleys in kspace and their first four commensurate orbits. __________________________ 35

5.3

Magnetoresistance data from all four antidot regions (sample M415B1). ______ 37

5.4

Magnetoresistance data from the a = 0.8 µm antidot region at various densities. ________________________________________________________ 38

5.5

Simulation snapshots showing various types of trajectories: chaotic, pinned and skipping orbits for both X and Y valley electrons. _______________ 40

5.6

Magnetoresistance obtained from numerical simulations. __________________ 41

5.7

Summary of the density dependence of the main commensurability peaks for all antidot regions. ______________________________________________ 43

5.8

Fourier analysis of the Shubnikov-de Haas oscillations from a = 0.8 µm antidot region to deduce the density imbalance in the system. _______________ 44

5.9

Summary of the density dependence of the two sets of commensurability peaks A and B for a = 0.6 µm and a = 0.8 µm antidot regions. ______________ 45

6.1

Schematic diagram of a split-gate quantum point contact (QPC) device and the original quantized conductance data in the QPC. ___________________ 47

6.2

Conductance vs. gate voltage in an AlAs QPC and its corresponding transconductance ( dG / dVG ) trace. ___________________________________ 48

6.3

Device schematic of the shallow-etched QPC. ___________________________ 50

6.4

Schematic of the potential landscape surrounding the QPC and the dependence of the Fermi energy on the gate voltage. ______________________ 53

6.5

Magnetoresistance traces for the AlAs QPC device._______________________ 55

6.6

Fourier spectrum of the longitudinal magnetoresistance and the differential of the transverse magnetoresistance ( dRxy / dB ) to deduce various density components in the 2D reservoir and the QPC. ______________ 56

6.7

QPC channel electrical width, deduced from the kink in the magnetoresistance. _______________________________________________________ 57

xi

6.8

A revised QPC energy level model with variable channel width, showing the expected crossings of the Fermi energy and the quantized levels in the QPC. ________________________________________________________ 59

6.9

Differential conductance G = dI / dVSD map of the QPC. __________________ 61

6.10 Full-width at half-maximum of the zero bias anomaly peak as a function of gate voltage. ___________________________________________________ 62 7.1

An idealized (conventional) piezoresistance effect due to strain-induced intervalley electron transfer in AlAs 2DES, a simple two-valley system. ______ 65

7.2

The experimental setup of the giant piezoresistance experiment in the AlAs 2DES with antidot lattices. _____________________________________ 67

7.3

The giant piezoresistance effect in an AlAs 2DES from both the blank and the antidot regions (sample M409K8). ______________________________ 69

7.4

The gauge factor vs. strain data calculated from Fig. 7.3.___________________ 70

7.5

Finite element simulation of the strain distribution in a 2D medium perforated with an antidot lattice. _____________________________________ 71

7.6

Shubnikov-de Haas oscillations of the blank region at various piezo bias (strain) values and their corresponding Fourier spectra (sample M409K8). _______________________________________________________ 74

7.7

Shubnikov-de Haas oscillations of the a = 0.6 µm antidot region at various piezo bias (strain) values and their corresponding Fourier spectra (sample M409K8)._________________________________________________ 75

7.8

The low-field magnetoresistance data from all antidot regions showing the commensurability peaks associated with a fundamental peak and their subharmonics (sample M409K8). ____________________________________ 76

7.9

The strain dependence of the fundamental commensurability peak at various gate voltages or densities (sample M409K8).______________________ 78

7.10 Magnetoresistance traces obtained from numerical simulations of the transport through antidot lattice with variable channel width that demonstrate the emergence of the sub-harmonic peaks. ___________________ 80 7.11 The strain (piezo bias) dependence of the low-field magnetoresistance for all antidot regions showing variation of the subharmonic peak amplitudes._______________________________________________________ 82 7.12 The variation of the commensurability peak amplitudes with strain (piezo bias) for the 1 µm-AD (a = 1 µm) region. ______________________________ 84 7.13 The analogy between the giant magnetoresistance effect in a layered magnetic metal sandwich structure and the giant piezoresistance effect in the AlAs 2DES with an antidot lattice. _________________________________ 85 xii

7.14 Density dependence of the piezoresistance for the 1 µm-AD region.__________ 87 7.15 Temperature dependence of the piezoresistance from the blank and the 1 µm-AD regions and their corresponding gauge factors. ____________________ 89 7.16 Testing the strain detection limit by modulating the piezo bias and monitoring the modulated resistance. __________________________________ 90 8.1

A more informative giant piezoresistance experiment in an AlAs 2DES with an antidot lattice in a Hall bar with van der Pauw geometry. ____________ 95

8.2

A schematic diagram for a valley filter device employing QPCs. ____________ 97

B.1 A specially designed Hall bar mask for AlAs 2DES devices with various new features. ____________________________________________________ 101 D.1 A single layer of piezoelectric element operating in the d33 mode and a typical “piezo stack” piezoelectric actuator. ____________________________ 108 D.2 A “T-Rosette” strain gauge used in this thesis. _________________________ 110 D.3 Experimental setup for the piezo strain factor (PSF) measurement. __________ 113 D.4 Calibration of the bridge circuit. _____________________________________ 115 D.5 Piezo bias modulation and the corresponding output from the bridge circuit showing a drift in the original signal.____________________________ 116 D.6 Temperature dependence of the PSF. _________________________________ 117 D.7 Strain gauge excitation current dependence of the PSF. ___________________ 118 D.8 Linearity between the piezo bias modulation amplitude and the bridge output signal (∆R). _______________________________________________ 119 D.9 Piezo modulation frequency dependence of the PSF. ____________________ 119 D.10 Piezo series resistance dependence of the PSF. _________________________ 120 E.1 A bipolar tunable voltage source circuit._______________________________ 122 E.2 An active voltage adder circuit.______________________________________ 123 E.3 A programmable high current source circuit. ___________________________ 124 E.4 A closed-loop temperature control of the Oxford 3He cryostat using the programmable current source and a software proportional-differential controller._______________________________________________________ 125 F.1

The stress situation in an antidot region showing the residual stress and the applied tunable stress components. ________________________________ 127

F.2

The Fermi seas and the bottom of the conduction bands of the nonuniformly strained antidot regions for X and Y valley electrons. ____________ 129 xiii

F.3

The channel-pinching effect seen in the X-valley Fermi sea with increasing applied stress. __________________________________________ 130

F.4

Numerical simulation of the channel-pinching effect. ____________________ 131

xiv

1 1CHAPTER Introduction

INTRODUCTION Our modern time has witnessed the birth and explosive growth of the electronics technology. It began with the invention of transistor that led to the development of microcomputers, ushering in the era of information technology. These technologies have tremendous impact to the world’s economic growth [1]. With pervasive influence and applications, electronic devices and instruments are indispensable, often critical in our everyday life. Needless to say, electronics technology has been the cornerstone of modern civilization [2]. At the heart of the immensely successful electronics technology is the ability to control and manipulate electron charge to process information. Starting from the bipolar junction transistor [3], a host of other electronic devices were developed such as the diode, field effect transistor (FET), thyristor, charge-coupled device, etc. In these devices one basically uses the electric field to control the electron charge. Following suit, many researchers recently started to look into another degree of freedom of electron, namely spin, marking the birth of a new field called spintronics, or spin-based electronics. In spintronics, the primary control over electron spin is achieved using magnetic field. There is yet another electronic degree of freedom in certain types of semiconductors that is relatively unexplored and may hold promise for future technology: valley occupation. Valley is a local minimum point in the conduction band structure of a semiconductor material where the electrons reside in k-space. Since the valley, or the band structure in general, originates from of the crystal structure of the material, it is sensitive to the deformation of the crystal. Therefore a means to control the valley occupation is by manipulating the strain field in the crystal. Following similar nomenclature, the research of valley-based electronics could be referred as “valleytronics”. This research perspective in terms of which electron property one manipulates is summarized in Fig. 1.1 below.

1

2

Notes:

CCD = charge coupled device MRAM = magnetic random access memory

GMR = giant magnetoresistance GPR = giant piezoresistance

Figure 1.1 Research perspective on the manipulation of electrons in solid-state devices based on the electrons’ properties such as charge, spin and valley degree of freedom (in a multivalley semiconductor). Ee, EZ and EV represent various energy scales namely, the electron’s (kinetic) energy, Zeeman splitting and the valley splitting respectively.

The study of the electronic valley degree of freedom is of great importance and relevance since after all Si, the most technologically important semiconductor, is a multi-valley system (for n-type). Furthermore, the conduction-band valley is essentially a quantum mechanical property of electrons in a solid, implying possible applications in the area of quantum computation where the valley index might be utilized as a qubit [4]. As a comparison, there have been intense efforts to utilize the spin degree of freedom to realize a solid-state quantum computer [5,6]. To certain extent, the valley degree of freedom has been exploited by CMOS manufacturers via a strain engineering technique to enhance the electrons’ mobility [7,8].1 However such a technique does not allow an in-situ control of the valley 1

Another example of device exploiting the valley degree of freedom is the Gunn diode [9]. However in this device, the carriers’ valley occupation is controlled by electric field through hot electron effect instead of strain field.

3

occupation thus hindering further exploration of the valley physics. This fact is further complicated by the large (sixfold) valley degeneracy present in n-type Si. This thesis presents a study of ballistic transport in a two-dimensional electron system (2DES) with a two-fold valley degeneracy, the simplest multi-valley system. The system is realized in a high-quality, wide AlAs quantum well (QW) [10,11]. The high mobility attained allows experimentation in the ballistic transport regime where the dimensions of the conducting channel are smaller than the electron mean-free-path so that electronic transport (at low temperatures) is dominated by boundary scattering rather than by scattering from impurities or phonons. We studied some basic properties of the ballistic transport unique to this two-valley system in several mesoscopic devices such as a surface-grating device, an antidot lattice device, and a quantum point contact. In Chapter 2 we review some general information on multi-valley systems and the basic properties of the 2DES confined to AlAs QWs. We discuss the valley degeneracies in bulk AlAs and in AlAs QWs and briefly describe the technique we use to tune the inplane valley populations in AlAs wide QWs. We also present a general description of ballistic transport. Chapter 3 deals with the experimental details such as the device fabrication and measurement techniques. We highlight some original ideas and developments which are particularly important for the AlAs 2DES device research. In Chapter 4 we report a ballistic transport experiment in a surface-grating device that exhibits commensurability oscillations in the magnetoresistance traces. We induce a onedimensional modulation potential in the 2DES with a surface-grating that leads to a geometric resonance effect in the presence of a perpendicular magnetic field. Here we demonstrate the first observation of a valley-resolved ballistic transport and, furthermore, deduce the effective mass anisotropy factor ml / mt where ml and mt are the longitudinal and the transverse effective masses respectively. Using inverse Fourier analysis we disentangle the two transport components arising from the two valleys. In Chapter 5 we present another type of ballistic transport experiment, i.e. transport in an antidot lattice. These experiments reveal another remarkable ballistic transport phenomenon associated with magnetoresistance peaks arising from the electron orbits becoming commensurate with the antidot lattice. From the analysis of the peaks

4

associated with the shortest commensurate orbits we deduce the mass anisotropy factor ml mt . Chapter 6 covers our study of an AlAs 2DES quantum point contact (QPC) device. We present a successful demonstration of quantized conductance in an AlAs QPC. Thanks to the large value of the effective mass, the subband levels in the QPC constriction are very closely spaced, making the observation of quantized conductance difficult. From our analysis we deduce that it is the valley with larger mass (i.e. the longitudinal mass, ml) along the QPC confinement potential that dominates the low-lying subband energies in the QPC. This suggests the potential use of the QPC as a natural “valley-filter”. As another interesting finding, we also observe a “0.7 structure” i.e., conductance quantized at ≈ 0.7 (2e 2 / h) , which is stronger than the other quantized plateaus, suggesting its different origin. In Chapter 7 we present a surprising finding, namely an anomalous giant piezoresistance effect in AlAs 2DESs with antidot lattices. We demonstrate that it is possible to engineer such devices to achieve very high and thermally stable piezoresistivity. Such a device could be utilized as an ultra sensitive strain sensor. In this chapter we also present extensive magnetoresistance data and a model that accounts for many of the features observed in the experiment. In Chapter 8 we summarize the work presented in this thesis and present a number of ideas to pursue in the future. Finally in Appendices A through F we document some additional information and important details of the experiments that have been originally developed in this thesis work. Appendix A contains a list of publications originating from this work. We then describe a new Hall bar design in Appendix B. Appendix C presents a summary of various AlAs effective mass determinations found in the literature. Appendix D describes an experimental technique to accurately calibrate the piezo strain factor. We also document some circuit diagrams for custom-made instruments that are useful for device characterization in Appendix E. To end with, in Appendix F we present an extended model that describes the anomalous giant piezoresistance effect discussed in Chapter 7 in more quantitative detail.

2 Section 2Chapter 2 2CHAPTER Equation

BACKGROUND This chapter presents some background information and basic ideas for the work presented in this thesis. We start with a discussion of multivalley semiconductors and then describe the realization and some basic properties of the two-dimensional electron system (2DES) confined to AlAs quantum wells (QWs). Finally we describe the ballistic transport regime which is explored in this thesis.

2.1 MULTIVALLEY SEMICONDUCTORS The single most important property of a semiconductor material is its energy band structure as it governs many of its electrical and optical properties. The band structure stems from a quantum mechanical description of the motion of electrons in the crystal. The minima in the conduction band (or “valleys”) determine where the electrons reside in the momentum space (k-space) and form “pockets” of electrons. Examples of multivalley solids are n-type Si, Ge, AlAs and PbTe. As an example, the band structure of AlAs is shown in Fig. 2.1 (a). It demonstrates an indirect bandgap, where the conduction band minimum is located at the X point of the Brillouin zone (BZ) (indicated by an arrow as X6), away from the maximum point of the valence band at the Γ point. Being away from the symmetric Γ point, the conduction band valleys in AlAs possess anisotropic constant energy surfaces, meaning that the electron effective masses are different along the longitudinal and transverse directions due to different energy dispersion curves. The constant energy surface is a prolate ellipsoid as shown in Fig. 2.1 (b). This surface can be described by the following equation (for the case of a valley along the [100] direction and centered at k0): 2 2 = 2  ( k x − k0 ) 2 k y kz  E=  + +  2  ml mt mt 

(2.1)

The equation indicates that the ellipsoid is characterized by two important parameters: the longitudinal (ml) and the transverse (mt) effective mass. The cubic crystal symmetry in AlAs dictates that the X point is six-fold degenerate in the first BZ as shown in

5

6

2.1 MULTIVALLEY SEMICONDUCTORS

Fig. 2.2 (a). However, for AlAs, the ellipsoid is right at the face of the BZ so effectively there are three full ellipsoids (six halves). We label these valleys as X, Y and Z valleys according to the direction of their principal axis: x, y and z. (a)

(b)

Figure 2.1 (a) Band structure of AlAs [12,13]. The arrow indicates the conduction band minimum at the X point of the Brillouin zone where the electrons reside. (b) The constant energy surface of one valley with its principal axis along [100].

ml = 1.1 m0

mt = 0.20 m0

ml = 0.92 m0

mt = 0.19 m0

ml = 1.64 m0

mt = 0.082 m0

gv = 3

gv = 6

gv = 4

(a)

(b)

(c)

Figure 2.2 Constant energy surfaces in k-space for the conduction band edge of three multivalley semiconductors: (a) AlAs, (b) Si and (c) Ge. The values of the longitudinal (ml), transverse effective masses (mt) and valley degeneracy (gv) are indicated.

2.2 ALAS TWO-DIMENSIONAL ELECTRON SYSTEM

7

For comparison, Fig. 2.2 also presents the constant energy surfaces of other technologically important semiconductors: Si and Ge. As we can see, AlAs is very similar to Si except that Si has six degenerate valleys lying along the ∆-line (Γ → X line) of the BZ, about 85% of the way to the zone boundary (X point). Ge has its conduction band minima at the L points of the BZ. Like AlAs, Ge’s valleys are located right at the face of the BZ, therefore Ge has a four-fold valley degeneracy. If the electrons populate only a single valley, the electrical properties are highly anisotropic. The electrons would have a high mobility in the direction where the effective mass is small, and a lower mobility where the effective mass is large. This property has been exploited to yield a large piezoresistance effect and can be utilized to realize a very sensitive strain sensor [14-16]. However in bulk multivalley material, the electrons in the whole set of valleys contribute to conduction and thus lead to an isotropic conductivity as a consequence of the cubic symmetry of the crystal.

2.2 ALAS TWO-DIMENSIONAL ELECTRON SYSTEM Modern crystal growth technology such as MBE (molecular beam epitaxy) has allowed the fabrication of high purity material and heterostructures. One important device structure is the MODFET (modulation-doped field-effect transistor), also known as HEMT (high electron mobility transistor). MODFET is a field-effect transistor device that typically has a 2DES trapped at a heterojunction interface, with dopants that provide modulation doping located at separate locations [See, e.g., Fig. 2.3]. While in the Si/SiO2 MOSFET system the highest electron mobility achieved is around 4 m2/Vs, in a MODFET one can achieve values over 1000 m2/Vs in GaAs 2DESs [17]. These are mobilities measured at low temperatures, where they are limited by scattering from impurities, defects and interfaces rather than phonons. The extremely high mobility in GaAs 2DESs is attributed to the almost perfect crystalline quality of the GaAs/AlGaAs heterostructures and the ability to separate carriers from the dopant impurities. Our research has concentrated on the growth and characterization of AlAs QW structures. Being closely related to GaAs, AlAs enjoys many advantages such as a lattice constant which is closely matched to GaAs; this allows a high quality and dislocation-free AlAs /AlGaAs interface. By adopting a single-sided doping structure, a record high electron mobility of 31 m2/Vs has been achieved [11].


8

Figure 2.3 The layer structure of AlAs MODFET wafers used in this thesis: (a) M415 (15 nm wide QW) and (b) M409 (11 nm wide QW). δ-Si indicates a delta-doped layer. (c) The energy band diagram of M409 [11] showing the conduction band edges at the X and Γ points of the Brillouin zone.

Figure 2.4 Valley occupation in AlAs QWs and the (in-plane) 2D Fermi contours for: (a) Narrow well (w < 55 Å) and (b) Wide well (w > 55 Å).


9

When we confine electrons to an AlAs QW, two mechanisms lift the valley degeneracy: the quantum confinement effect of the QW, and the strain effect arising from the lattice mismatch between GaAs and AlAs. The quantum confinement causes the valley with larger mass along the confinement direction, i.e. the out-of-plane (Z) valley, to have lower energy. On the other hand, the slightly larger lattice constant of AlAs ( a AlAs = 5.6611 Å) compared to the GaAs ( aGaAs = 5.6533 Å) leads to a biaxial compressive strain that lowers1 the energy of the in-plane valleys (X and Y) thus favoring them to be the ground state [10,18-23]. These two effects compete with each other and, depending on the thickness (w) of the QW, we have two cases of valley occupation in AlAs QWs [10,18-23]: 1. Narrow AlAs QW (w < 55 Å): The confinement effect dominates and the out-of-plane valley (Z) becomes the ground state as shown in Fig. 2.4 (a). 2. Wide AlAs QW (w > 55 Å): The strain effect dominates and the in-plane valleys (X and Y) become the ground state as shown in Fig. 2.4 (b). This thesis focuses on the wide AlAs QWs where the 2DES occupies the X and Y valleys. This system has some unique properties that are very different from those of the more commonly studied GaAs 2DES [11,24]: large and anisotropic effective mass, large Landé g-factor and, most importantly, the possibility to tune the valley populations. The crystal and band structures of the two wafers that we used in the experiments reported in this thesis are given in Fig. 2.3. We have discussed how the strain due to lattice mismatch breaks the degeneracy between the Z and the X and Y valleys in AlAs QWs. One can further lift the degeneracy between the X and Y valleys by applying symmetry-breaking strain along the [100] or [010] directions [25]. This is achieved by gluing the sample on top of a piezo-actuator [Fig. 2.5 (a)] and thus controlling the valley splitting that induces transfer of electrons from one valley to the other as shown in Figs. 2.5 (b) and (c). This experimental setup

1

By convention, compressive strain has negative value and lowers the valley energy.

10


(a)

(b)

(c)

Figure 2.5 (a) Experimental setup for tuning the valley populations in AlAs 2DES using a piezo-actuator. (b) Electron transfer from the X to the Y valley occurs with the application of positive piezo bias (VP) to dilate the sample along [100] and shrink it along [010]. (c) Energy diagram showing that the valley energies and populations are split by the symmetry-breaking strain.

is utilized in Chapter 7 to demonstrate an anomalous giant piezoresistance effect in an AlAs 2DES with an antidot lattice. In the experimental setup as shown in Fig. 2.5 (a), the valley splitting ∆EV between the X and Y valleys is given as: ∆EV = EV , X − EV ,Y = E2 (ε[100] − ε[010] ) ,

(2.2)

∆EV = E2 ε ,

(2.3)

where E2 = (5.8 ± 0.1) eV is the AlAs shear deformation potential [26] and ε = ε[100] − ε[010] is the symmetry-breaking strain (heretofore simply referred to as strain). In the case of the piezo-actuated device as shown in Fig. 2.5 (a), at low temperature this strain is proportional to the applied piezo bias, VP : ∆ε = κ ∆VP

(2.4)

where κ is the piezo strain factor.2 The value κ can be determined using the strain gauge glued to the other side of the piezo [Fig. 2.5 (a)]. We have developed a reliable technique to measure κ at low temperatures as described in Appendix D. We would like to emphasize that among all multivalley semiconductor systems, the 2

Near room temperature the strain vs. piezo voltage exhibits significant non-linearity and hysteresis but not at or below the liquid He temperature [25].

2.3 BALLISTIC TRANSPORT

11

opportunity to have a strain tunable two-valley system is unique to the AlAs wide QWs,3 thanks to the slightly larger AlAs lattice constant compared to GaAs that leads to biaxial compressive strain.4 This valley-tunability is a critical feature that allows us to unlock a wealth of phenomena related to the role of the valley degree of freedom in 2DESs. Among recent findings in our group are: spin susceptibility dependence on the valley degree of freedom [28], observation of valley skyrmions [29], giant piezoresistance effect [16], enhanced valley susceptibility values [4], spin-valley phase diagram of the twodimensional metal-insulator transition [30], and parallel field induced valley imbalance [27]. A summary of some of these results can be found in Ref. [31].

2.3 BALLISTIC TRANSPORT Electrical transport phenomena can be broadly categorized into two different regimes based on the relative size of the carrier’s elastic mean free path (le) and the relevant feature sizes of the sample such as width (W) or length (L). The elastic mean free path is the distance over which the carrier can travel without experiencing elastic scattering so that its momentum and energy are conserved. At low temperature and low bias excitation, where the current is carried only by the electrons at the Fermi energy, the mean free path is given by: le = vFτ e , where vF is the Fermi velocity and τ e is the elastic scattering time. If the mean free path is much smaller than the relevant dimensions of the sample, electron transport is in the diffusive regime. In this regime transport is dominated by the usual scattering processes such as impurity, alloy or phonon scattering, as shown in Fig. 2.6 (a). Transport is basically Ohmic and the usual definition of resistance applies. However, when we shrink our device size or if we improve the mobility of the carriers, the elastic mean free path becomes comparable or larger than the relevant dimensions of the sample and transport is in the ballistic regime. Most of the scattering is now dominated by boundary scattering as shown in Fig. 2.6 (b). The exploration of the ballistic transport regime in semiconductors has become routine in recent years thanks to the realization of high mobility 2DESs in modulation-doped GaAs/AlGaAs heterostructures (see, e.g.,

3 4

Strong perpendicular [24] or parallel [27] magnetic fields could also lift the valley degeneracy. In contrast, in Si1-xGex/Si QWs for example, the lattice constant of the Si layer (the QW layer) is smaller than Si1-xGex, thus the Si layer experiences an in-plane tensile strain, lifting the in-plane valley energies. The confinement further enhances this lifting and, as a result, only the two out-of-plane valleys are occupied.

2.3 BALLISTIC TRANSPORT

12

Figure 2.6 Electron trajectories in the two transport regimes: (a) diffusive and (b) ballistic.

Ref. [17]). Such studies have led to the discovery of several new phenomena such as magnetic focusing [32], commensurability oscillations in lateral superlattice devices [33], and quantized conductance in a quantum point contact [34]. The elastic mean free path for a 2DES is given by: le µ = 2πn / e where µ is the electron mobility and n is the electron density. For example, in our typical AlAs 2DES at T 0.3 K with a total electron density of nT 6×1011 /cm2 and a corresponding mobility of µ 10 m2/Vs [11], we have an elastic mean free path of le 1 µm.5

In this thesis we describe various ballistic transport experiments in the AlAs 2DES: commensurability oscillations in surface-grating devices, commensurability peaks in antidot lattices, and quantized conductance in a quantum point contact. In these experiments the relevant feature sizes of the sample are comparable to or smaller than the elastic mean free path. These feature sizes are the grating period in the commensurability oscillations experiment, the period of the antidot lattice, and the channel length (and also width) of the quantum point contact constriction. 5

In calculating the mean free path we use n = nT / 2 to account for the valley degeneracy in the AlAs 2DES and a circular (instead of elliptical) Fermi contour as an approximation.

3CHAPTER Chapter33

EXPERIMENTAL DETAILS In this chapter we describe the main experimental procedures: the sample fabrication process and the measurement techniques. We highlight some new techniques and improvements that have been developed in this thesis work.

3.1 SAMPLE FABRICATION

Figure 3.1 The main steps of the AlAs device fabrication process.

Figure 3.1 presents the main steps of the AlAs device fabrication process. A more detailed, step-by-step account is documented in Appendix B of Ref. [23]. General information on GaAs material processing techniques can be found in a book by R. Williams [35]. We detail the procedures used in each step of Fig. 3.1 as follows:

13


14

1. Sample cleaving We cleave samples from the MBE-grown wafer into square shapes of 4×4 mm2 to 5×5 mm2 to fit the arrangement of the Ohmic contacts dictated by a prefabricated shadow mask. Since the sample is grown on a GaAs (001) substrate, the principal crystal axes [100] and [010] lie along the diagonals of the cleaved piece. After cleaving, we clean the sample using acetone and methanol. 2. Ohmic contact deposition The quality of AlAs Ohmic contacts depends critically on the cleanliness of the surface prior to contact metal deposition. Remnants of photoresist could easily ruin the Ohmic contacts; therefore, we normally deposit the Ohmic contact materials first using a shadow mask prior to Hall bar patterning that uses photoresist.1 The shadow masks are fabricated from thin (0.4 mm thick) G10 composite plastic, instead of Al which was used previously. The G10 composite is chosen for its semi-transparent property that makes it easy to align to the sample. We coat the G10 shadow mask with poly-methyl-metacrylate (PMMA) prior to contact evaporation so that it can be re-used by lifting off the PMMA (and the metal on top) with acetone afterwards. The G10 plastic is quite strong and resistant to acetone. The Ohmic contact alloy consists of Au, Ge, Ni and Au with thicknesses of 20, 40, 10 and 40 nm respectively in order of the deposition steps. The alloy is then annealed in a forming gas (10% H2 and 90% N2) environment at 470 °C for 11 minutes to allow it to diffuse into the sample and make contact to the electrons in the quantum well (QW). 3. Hall bar patterning The Hall bar mesa defines the device and contact terminals for longitudinal and transverse resistance measurements typical in a quantum Hall experiment. We designed a new set of Hall bar masks suited for AlAs devices and electron beam lithography (EBL) process that incorporates alignment marks, EBL focusing pads, and multiple Hall bar regions for redundancy. These features are critical to achieve a successful and productive experiment. A complete description of this

1

Recently it was found that it is possible to pattern AlAs Ohmic contacts using a standard photolithography technique. To ensure the complete removal of photoresist and a clean surface, one can use a bilayer photoresist and a metal-ion free developer [36].


15

Hall bar design is given in Appendix B. Hall bar mesas are defined on the sample using a standard UV photolithography technique. We use a GaAs resist primer, Surpass 3000, prior to photoresist deposition to enhance adhesion of the photoresist to the substrate; this leads to a more faithful pattern transfer. The Hall bar is aligned along the [100] or [010] direction so that the major axes of the two in-plane valleys are either parallel or perpendicular to the Hall bar. For AlAs this means that the Hall bar has to be oriented along the diagonal direction as shown in Fig. 3.1 We wet-etch the mesa using a H3PO4:H2O2:H2O solution with a ratio of 1:1:40 that gives a fast etching rate of 400 nm/min. We typically etch the mesa very deep, down to 300 nm. (The QW is located at ~100 nm below the surface.) This is done to produce a high contrast image of the mesa edges during the EBL step. Typically a semiconductor surface is difficult to view using the scanning electron microscope (SEM) due to a poor image contrast. After etching we strip the photoresist completely using acetone. 4. Mesoscopic pattern fabrication Most of the samples in this thesis contain mesoscopic patterns such as surfacegrating, antidot lattice, or quantum point contact. Since their feature sizes are smaller than 1 µm, they have to be defined by EBL and, if necessary, followed by wet or dry etching. Electron beam lithography

We use 2.4% PMMA dissolved in chlorobenzene solution as a resist. The resist is deposited using a standard spin-coating technique at 8000 rpm spinning speed to achieve a film thickness of ~600 nm. The patterns are designed using standard computer aided design (CAD) programs such as CorelDraw, AutoCAD or DesignCAD. The EBL system is a JEOL 840, a modified SEM to perform lithography. The control program is the Nanometer Pattern Generation System (NPGS) from J. C. Nabity Lithography Systems. This program can perform software pattern-alignment, pattern-writing, as well as image acquisition. To write the pattern, the CAD design is first compiled into the machine code by specifying the writing parameters such as dosage, current and magnification settings. The software then executes the program to control the electron beam


16

position and dwell-time. After the lithography step, we develop the pattern for ~50 sec in methyl-isobutyl-ketone (MIBK) and isopropanol (IPAL) with a volumetric ratio of 1:3. Prior to writing on the real sample, we usually first test the lithography process outcome by writing to dummy samples with varying dosages. Optimum pattern demands correct dosage, good focusing, and appropriate beam current. Wet etching

After pattern definition by EBL, one may need to perform etching to define the pattern. A simple option is to do wet etching using a standard acid solution. The advantages are simplicity, low cost, reasonable accuracy and reproducibility in etching depth, and more importantly, less damage induced to the sample in contrast to the dry etching technique. However, one major disadvantage is that the wet etching technique cannot be used to etch very small patterns because of the hydrophobic nature of the PMMA that repels the etchant. In this thesis, wet etching is used to define quantum point contact constrictions (Chapter 6) where we use a slow etchant: H2SO4:H2O2:H2O with a volumetric ratio of 1:8:160 that gives an etch rate of 240 nm/min. Dry etching

Dry etching, especially for GaAs materials, is an important processing step for the fabrication of high-speed electronic and optoelectronics devices. In this thesis, we use reactive ion etching (RIE) with a high density electron cyclotron resonance (ECR) plasma, often referred to as the ECR-RIE process [37,38]. The ECR-RIE technique performs etching using a high plasma density, low-pressure, and low-temperature environment [39,40]. The high plasma density allows for a low energy operation that introduces little damage to the sample. The ECR plasma is created by a combination of absorption of microwave radiation and a magnetic field that induces the electron cyclotron resonance at low gas pressure. The substrate holder is negatively biased to attract the positively-charged plasma that creates a bombardment of the substrate surface by ions and free radicals. This bombardment provides both chemical and physical etching of the sample surface.

17


(a)

(b)

Figure 3.2 (a) Schematic diagram of the Astex ECR-RIE system. (b) The test pattern used in a routine process calibration prior to every etching session.

We use an Astex ECR-RIE system whose schematic is shown in Fig. 3.2. The main components are a 2.45 GHz microwave source, magnetic coils (that provide a magnetic field of ~8.8 mT to induce the cyclotron resonance), an RFpowered substrate holder built for 4-inch wafers, and a vacuum reactor chamber connected to a turbo molecular pump and gas inlets. The ECR-RIE technique has broad process windows and a variety of applications. The advantages are a highly anisotropic etching process, relatively low damage (to the electronic properties of the 2DES) due to its low energy compared to other RIE techniques, and the ability to etch small features ( 1.7 T). Inset: numerically determined second derivative d 2 ρ xx / dB 2 . (b) Fourier power spectra of COs from both ρ xx (solid curve) and d 2ρ xx / dB 2 (dotted curve). (c) Fourier power spectrum of SdHOs.

4.3 EXPERIMENTAL RESULTS

27

Figure 4.4 CO traces from M409N3: (a) Series of low field MR traces exhibiting COs at various densities. The traces are offset for clarity. (b) The corresponding Fourier power spectra for the original (solid line) and the second derivative (dotted line) of the CO signal in the range 0.1 T to 1 T. The dashed line is a guide to the eye.

The SdHOs provide information regarding the electron densities of the 2DES and the valleys. In Fig. 4.3 (c) we show the Fourier power spectrum of the SdHOs. To calculate this spectrum, we used the ρ xx vs. 1/B data for B >1.7 T, subtracted a second-order polynomial background, and multiplied the data by a Hamming window [46] in order to reduce the side-lobes in the spectrum. The spectrum exhibits three peaks, marked in Fig. 4.3 (c) as nT, nT/2 and nT/4. The peak frequencies multiplied by e / h give the 2D density (e is the electron charge and h is Planck's constant). We associate the nT peak with the total density, as this peak's frequency multiplied by e / h indeed gives the total 2DES density which we independently determined from the Hall coefficient. For Fig. 4.3 data, we deduce nT = nX + nY =

28

4.4 ANALYSES AND DISCUSSIONS

8.7×1011 /cm2. The presence of the nT/4 peak indicates the spin and valley degeneracy of the 2DES.4 Figure 4.3 (b) shows the Fourier power spectra of COs calculated using ρ xx and d 2 ρ xx / dB 2 vs. 1/B data in the 0.1 < B < 1 T range. Both spectra exhibit two clear peaks at fCO,X and fCO,Y, which we associate with the CO frequencies of the X and Y valleys, respectively. If we assume that the two valleys have equal densities, we can use Eq. (4.6) to immediately find ml / mt = 4.4 . This value, however, is inaccurate because there is a small but finite imbalance between the X and Y valley densities in our sample. Such imbalances can occur because of anisotropic strain in the plane of the sample and are often present in AlAs 2DESs. Note that the Fourier spectrum of the SdHOs cannot resolve small valley density imbalances. Figure 4.4 presents several CO traces and their Fourier spectra at different densities achieved by varying the top gate bias. As detailed in the next paragraph, we analyze the dependence of CO frequencies on density to deduce the imbalance between the valley densities, and also to determine the ml / mt ratio more accurately.

4.4 ANALYSES AND DISCUSSIONS 4.4.1 Determination of Mass Anisotropy Ratio ml / mt Figure 4.5 summarizes the density dependence of our CO frequencies. Denoting the difference between the valley densities by ∆n = nY − n X , we rewrite Eqs. (4.3) to (4.5): f CO2 ,Y =

h π e2 a 2

ml ( nT + ∆n ) , mt

(4.7)

f CO2 , X =

h π e2 a 2

mt ( nT − ∆n ) . ml

(4.8)

The slopes and intercepts of the f CO2 vs. nT plots give the ml / mt ratio and ∆n . Concentrating on the Y valley, a least-squares fit of f CO2 ,Y data points (circles in Fig. 4.5) to a line leads to values ml / mt = 5.2 ± 0.5 and ∆n = (−0.6 ± 0.4) × 1011 cm −2 . Note that

4

As detailed in Ref. [24], the spin and valley degeneracies are lifted at higher B, leading to the presence of the nT/2 and nT peaks in the Fourier spectrum.


29

Figure 4.5 Density dependence of the CO frequencies for the Y (circles) and X valleys (squares) for sample M409N3 (a = 400 nm). The line through the circles is a least-squares fit to the data; its slope determines the ratio ml / mt and its intercept the density difference ∆n of the two valleys. The dashed line is described in the text.

such a small value of ∆n is consistent with the nearly valley-degenerate picture deduced from the existence of the nT / 4 peak in the SdH frequency spectrum [Fig. 4.3 (c)]. The above determination of the ml / mt ratio is based on the density dependence of f CO ,Y only and does not use the measured f CO , X . As a consistency check, we use Eq. (4.8) to predict f CO for the X valley using ml / mt and ∆n deduced from the above analysis of f CO2 ,Y above. This prediction, shown as a dashed line in Fig. 4.5, agrees well with the measured f CO2 ,Y (solid squares), and confirms that we are indeed observing COs for both valleys. We repeated similar experiments in a sample (M415L3) from a different wafer, containing a 2DES confined to a 15 nm wide AlAs quantum well. The data for this sample are summarized in Fig. 4.6. In this sample only the COs of the Y valley could be reliably determined. By performing similar analysis using Eq. (4.7), in the density range


30

Figure 4.6 CO data for M415L3 (a = 300 nm): (a) Density dependence of the CO frequencies, similar to Fig. 4.5. (b) The CO trace at n = 6.4×1011 /cm2. The trace in red is the second derivative d 2ρ xx / dB 2 . (c) The corresponding Fourier power spectrum of the CO taken from ρ xx (black) and d 2ρ xx / dB 2 (red).

from 5 to 8.5×1011 cm-2, we deduce ml / mt = 5.4 ± 0.5 , in good agreement with the results for M409N3. At this point it is worthwhile emphasizing that the COs described here uniquely probe the ml / mt ratio.5 Conventional experiments that probe the effective mass, such as cyclotron resonance or measurements of the temperature dependence of the amplitude of the SdHOs, lead to a determination of the cyclotron effective mass, mCR . In a 2DES with an elliptical Fermi contour, mCR is equal to ml mt , and therefore provides information complimentary to the ml / mt ratio, so that ml and mt can be determined. In fact, using the measured mCR = 0.46me in AlAs 2DESs,6 we use the ml / mt = 5.2 ± 0.5 ratio to 5

6

Faraday rotation experiments can also determine the ml / mt ratio, but such determination requires 2 2 knowing ml or mt and the 〈τ 〉 / 〈τ 〉 ratio where τ is the scattering time. In fact, B. Rheinländer et al. [47] used Faraday rotation measurements in bulk AlAs and, assuming mt = 0.19 m0 (determined from a k ⋅ p 2 2 calculation) and 〈τ 〉 / 〈τ 〉 = 1 , deduced a ratio ml / mt = 5.7 . The most accurate cyclotron resonance (CR) measurements in AlAs 2DESs so far were reported by T. S. Lay et al. [10], and yielded mCR = (0.46 ± 0.02) me . This value is in very good agreement with the results


31

deduce ml = (1.1 ± 0.1) me and mt = ( 0.2 ± 0.02 ) me . These values are in good agreement with the (theoretical) value of mt = 0.19me that is calculated in Ref. [47] and ml = 1.1me that is deduced from the Faraday rotation measurements [47];5 they also agree well with the results of the majority of theoretical and experimental determinations of the effective mass in AlAs. A summary and discussion of AlAs effective masses can be found in Ref. [50] and is also summarized in Appendix C.

4.4.2 Resolving the Ballistic Transport in Individual Valleys We proceed to extract more information, such as the amplitude, phase, and scattering time from the COs of each valley by performing partial inverse Fourier analysis. Figure 4.7 summarizes the results of such analysis. The Fourier power spectrum shown in Fig. 4.3 (b) is separated into two ranges7 chosen to isolate the two CO peaks. The range for COs of Y valley (0.57 < fCO < 1.21 T) is inverse Fourier transformed and divided by the original window function. The result is shown as the solid curve in Fig. 4.7 (a). The range for the COs of X valley (0.29 < fCO < 0.57 T) is analyzed in a similar manner and the result is the solid curve in Fig. 4.7 (b). We fit the deduced COs for each valley to a simple expression that assumes the amplitude of the COs decreases exponentially with 1/ B : ∆ρ xx ∝ ρ0 exp(− π / ωCτ CO ) cos(2π f CO / B − θ 0 ) ,

(4.9)

where ρ0 ,τ CO , f CO , and θ 0 are the fitting parameters; ωC = eB / mCR is the cyclotron frequency with mCR = ml mt = 0.46me . The exponential term in Eq. (4.9) is analogous to the Dingle factor used to describe the damping of the SdHOs' amplitude with increasing 1/ B , and has been used successfully to fit COs in GaAs 2D electrons [51] and holes [52]. In Fig. 4.7 the results of the best fits are shown as dotted curves along with their fitting parameters. The best-fit θ 0 for the COs of Y and X valleys are 0.37π and 0.47π respectively, in excellent agreement with the expected value of 0.5π (the relative phase errors are 6 % and 2 % of 2π). This consistency affirms that the reconstructed oscillations

of CR measurements by N. Miura et al. [48] on n-type AlAs layers mCR = (0.47 ± 0.01) me , and by T. P. Smith III et al. [19] on 2DESs in multiple AlAs quantum wells ( mCR ~ 0.5 me ) ; the latter data, however, show a very broad CR. There was also a CR study of GaAs/AlAs short-period superlattices by H. Momose et al. [49], where ml = 1.04 me and mt = 0.21 me were deduced. 7

Varying these ranges by reasonable amounts (±10 %) does not lead to significant changes in parameters that are deduced from the inverse Fourier transform curves.


32

Figure 4.7 Results of the inverse Fourier decomposition of the COs of Fig. 4.3 for the Y and X valleys. The dotted curves show the best fits of Eq. (4.9) using the indicated parameters. The fits are done only for fields smaller than those marked by square points which indicate the positions of the first CO resonant orbits. The arrows indicate where the first two CO resonances occur.

faithfully represent the COs of the two valleys. The amplitude of the oscillations for the Y valley is larger than for the X valley as expected from the shorter real-space, resonant orbital trajectories for this valley [Fig. 4.2 (b)]. On the other hand, the scattering times, τ CO , that we deduce from the fits are comparable for the two valleys, suggesting that scattering is nearly isotropic.8 We also deduce two other scattering times: the quantum lifetime τ SdH and the mobility scattering time τ µ , and compare them with τ CO . From fitting the B dependence of the amplitude of the SdHOs in the patterned region to the damping factor exp( −π/ωcτ ) , we 8

For a circular cyclotron orbit trajectory, or for an elliptical orbit if we use the average Fermi velocity along the trajectory, the exp( − π/ω Cτ ) term in Eq. (4.9) is equivalent to exp( − L / 2l ) where L is the orbit length and l the mean-free-path. For the data of Fig. 4.7, a τ ~ 10 ps corresponds to l ~ 0.5 µm.

4.5 SUMMARY

33

obtain τ SdH = 0.76ps . The mobility scattering time is τ µ = 24 ps , determined from the mobility of the same sample prior to patterning. Similar to CO experiments in other 2D carrier systems [51,53], we observe τ SdH < τ CO < τ µ . This observation can be qualitatively understood considering the sensitivity of these τ to the scattering angle [51]: τ µ is the longest since the mobility is least sensitive to small-angle scattering, while τ SdH is the shortest because the SdHOs are sensitive to all scattering events (both small- and largeangle).

4.5 SUMMARY In summary, we have demonstrated valley-resolved ballistic transport in an AlAs 2DES using a surface grating device that introduces a lateral, one-dimensional, periodic modulation potential in the 2DES. We observe COs in the MR traces. The Fourier spectra of the oscillations reveal two distinct peaks associated with the transport from the two valley components. Using partial inverse Fourier analyses of the oscillations, we disentangle and study the COs of the electrons in the two valleys. Furthermore, this experiment allows us to probe the Fermi contour and deduce the effective mass anisotropy ratio ml / mt = 5.2 ± 0.5 .

5 in an Antidot Lattice 5CHAPTER AlAs 2DES

ALAS 2D ELECTRONS IN AN ANTIDOT LATTICE We describe in this chapter ballistic transport experiments on the AlAs two-dimensional electron system (2DES) in the presence of an antidot (AD) lattice where we observe peaks in the low-field magnetoresistance (MR). We present numerical simulations to elucidate the transport in this system and explain the resulting MR peaks. Similar to the commensurability oscillations experiments of the previous chapter, the data also provide a direct measure of the anisotropy of the Fermi contour, or equivalently ml / mt , the ratio of the longitudinal and transverse electron effective masses.

5.1 INTRODUCTION The commensurability oscillations presented in the previous chapter essentially demonstrate a geometrical resonance effect of the electrons’ cyclotron motion with a onedimensional periodic potential, i.e. a surface grating, that can be observed in transport measurements. Extending our study of this phenomenon, we perform similar experiments on samples with a two-dimensional periodic potential, i.e. an AD lattice. An AD lattice is a periodic array of holes, typically defined by etching. If imposed on a 2DES, the AD holes completely exclude the electrons thus creating a lateral superlattice in the 2DES. Such a structure alters the transport properties significantly and exhibits distinctive features in the MR. The system allows studies of classical chaos dynamics in condensed matter physics [54,55]. Early experiments in GaAs 2DESs with square (isotropic) AD lattices showed pronounced peaks in the low-field MR traces [56]. These peaks have been attributed to the pinned orbits around groups of ADs [56,57], as shown in Fig. 5.1, as well as runaway trajectories that skip from one AD to another [58,59]. These effects occur when the cyclotron diameter is commensurate with the lattice period. For example, in the GaAs 2DES where the Fermi contour is circular, the MR peak of the first commensurate orbit is observed at 2 RC = a , where RC is the radius of the cyclotron orbit and a is the AD lattice period.

34

5.1 INTRODUCTION

35

Figure 5.1 The results of the original AD lattice experiment by Weiss et al. [56] where the MR exhibits peaks associated with the commensurate orbits.

Figure 5.2 The AD lattice experiment in AlAs 2DES: (a) Micrograph of the AD lattice region with period a = 0.8 µm. (b) The Fermi contours of AlAs in-plane valleys X and Y in kspace. The Fermi wave vectors k F , X and k F ,Y are indicated. (c) The first four commensurate orbits for the X and Y valleys that give rise to peaks in MR for current along the x-direction; these orbits have diameters in the y-direction that are equal to a multiple integer of the AD lattice period (see text); this integer is given by the index i in Xi and Yi.

5.2 DEVICE FABRICATION

36

Here we report a similar experiment in the AlAs 2DES. The system provides a rather unique situation where we have a 2DES with anisotropic (elliptical) Fermi contours [Fig. 5.2 (b)] in an isotropic AD lattice as shown in Fig. 5.2 (a). We also perform an analysis of the MR peaks associated with the shortest commensurate orbit to determine the mass anisotropy factor ml / mt . To highlight the significance of this measurement, it is shown in the next chapter that this mass anisotropy can be exploited to realize a simple "valleyfilter" device using a quantum point contact structure [60]. Such a device may play an important role in "valleytronics" or valley-based electronics applications [61], or for quantum computation where the valley state of an electron might be utilized as a qubit [4].

5.2 DEVICE FABRICATION We performed experiments on 2DESs confined in a 15 nm-wide AlAs quantum well (sample M415B1) whose structure is shown in Fig. 2.3 (a) in Chapter 2. We patterned a Hall bar sample, with the current direction along [100], using standard optical photolithography. We then deposited a layer of PMMA and patterned the AD arrays using electron beam lithography. The PMMA layer served as a resist for a subsequent dry etching process used to define the AD holes. We used an electron cyclotron resonance etching system (see Chapter 3) with an Ar/Cl2 plasma [41], at an etch rate of ~55 nm/min, to obtain small feature sizes without a degradation of the 2DES quality. The AD pattern was etched to a depth of 80 nm, thus stripping the dopant layer and depleting the electrons in the AD regions. The micrograph of a section of one of our AD arrays is shown in Fig. 5.2 (a). Each AD array is a square lattice and covers a 20 µm × 30 µm area. There are four regions of AD lattice with different lattice periods: a = 0.6, 0.8, 1.0 and 1.5 µm as schematically shown in the inset of Fig. 5.3. The aspect ratio d / a of each AD cell is ~1:3, where d is the AD diameters. Finally, we deposited a front gate, covering the entire surface of the active regions of the sample to control the 2DES density. Following an initial back-gate biasing and brief illumination [42], we used the front gate to tune the total density (nT) from 2 to 5×1011 /cm2. This density was determined from both Shubnikov-de Haas oscillations and Hall coefficient measurements that agree with each other. From measurements on an unpatterned Hall bar region in a different sample but from the same wafer, we obtain a mobility of ~10 m2/Vs at a typical density of 3×1011 /cm2 and T = 0.3 K. This gives a typical mean-free-path of ~1 µm.


37

5.3 EXPERIMENTAL RESULTS Figures 5.3 and 5.4 summarize our main experimental results. Figure 5.3 shows the lowfield MR traces, measured as a function of perpendicular magnetic field (B), for all the AD regions. We observe two peaks, A and B, which are symmetric with respect to B = 0 T. Peak A, whose position is higher in field than peak B, is seen in the traces from all the AD regions. Peak B, on the other hand, is not observed in the a = 1.5 µm trace. In general, we observe that, as the AD lattice period becomes smaller, the positions of both peaks A and B shift to higher field values (as indicated by the dashed lines). Figure 5.4 captures the gate-voltage (VG ) dependence of the MR traces for the a = 0.8 µm AD region. As we increase VG to increase the 2DES density, peak A shifts to higher field values while peak B does not appear to shift. We have made similar observations in other AD regions as VG is varied.

Figure 5.3 Low-field MR traces for all four AD regions (sample M415B1) with periods equal to a = 0.6, 0.8, 1.0 and 1.5 µm (from top to bottom). The Hall bar with the different AD regions is schematically shown on the right. For clarity, traces are shifted down (from top to bottom) by: 2885, 390, 350, and 0 Ω.


38

Figure 5.4 Low-field MR traces (sample M415B1) for the AD region with period a = 0.8 µm for VG = -0.1 V to 0.15 V (from top to bottom), corresponding to a linear variation of the density nT from 2.27 to 3.53×1011 /cm2. For clarity, traces are shifted down (from top to bottom) by: 1410, 980, 645, 385, 175 and 0 Ω.

In order to analyze and understand the data of Figs. 5.3 and 5.4, we first briefly review what is known about ballistic transport in AD arrays for GaAs 2DESs where the Fermi contour is isotropic. Low-field MR traces for such systems typically exhibit commensurability peaks at magnetic fields where the classical cyclotron orbit fits around a group of ADs [54,56,62]. Although there are subtleties associated with the exact shape of the AD potential and also the possibility of chaotic orbits that bounce from one AD boundary to another, the peak observed at the highest magnetic field corresponds to the shortest period that fits around the smallest number of ADs; for an isotropic Fermi contour, this would correspond to a circular orbit, with a diameter equal to the AD period, encircling a single AD. There have also been studies of ballistic transport in 2DESs with

39

5.4 NUMERICAL SIMULATION

isotropic orbits in an anisotropic (rectangular) AD lattice.1 Experimental results [63], followed by theoretical analysis [64], have indicated that the commensurability peaks are observed only when the orbit diameter matches an integer multiple of AD lattice period along the direction perpendicular to the current.2 Based on the above considerations, we can predict the first four (smallest) commensurate orbits of the X and Y valleys that may give rise to MR peaks in our system; these are shown in Fig. 5.2 (c). To evince this conjecture we performed numerical simulations as described in the following section.

5.4 NUMERICAL SIMULATION To elucidate the transport mechanism in our samples we performed a kinematic, numerical simulation for our system, a 2DES with elliptical Fermi contours in an isotropic AD lattice in the presence of a perpendicular magnetic field B. We simulate the kinematic of a large number of electrons and calculate the MR based on a classical linear response theory using the Kubo formula [54,67]. The simulation details are similar to those described in Ref. [68]. We calculate two separate cases: X and Y-valley electron transport by assuming equal electron densities in each case. The kinematic of the electrons is governed by the following equations: x = vx , vx = −

eB vy , mx

y = v y , v y =

eB vx , my

(5.10) (5.11)

where mx and my are the effective masses along the x-direction and y-direction. The electrons are constrained to the elliptical constant energy (Fermi) contours. The Fermi velocity of the electron varies depending on its location on the Fermi contour. We use a large number of electrons (NP), typically 10,000, and have verified that the calculated resistances converge to within ±3.5 % of the asymptotic value for NP > 5,000. The AD boundaries are represented by hard-wall potentials so that the electrons are scattered elastically upon collision. The geometry of the AD lattice is based on the experimental parameter i.e. d / a = 1/ 3 . Figure 5.5 shows snapshots of the simulation at magnetic fields when the commensurate orbits X1 and Y1 occur (peaks X1 and Y1 in Fig. 5.6). 1

2

The problem of ballistic transport for a 2DES with a circular Fermi contour in a rectangular AD lattice is equivalent to transport in a 2DES with an elliptical Fermi contour in a square AD lattice. This situation resembles the magnetic electron focusing effect in a system containing multiple, parallel one-dimensional channels. See, e.g., Ref. [65] and [66].

40


Figure 5.5 Simulation snapshots showing various types of trajectories: (i) chaotic, (ii) pinned and (iii) skipping orbits for: (a) X-valley electrons at B = 4 ml / mt B0 (X1 orbit), and (b) Yvalley electrons at B = 4 mt / ml B0 (Y1 orbit). B0 is the magnetic field of the first commensurate orbit if the Fermi contour were circular.

The conductivity of the system can be calculated based on classical linear response theory where the Ohmic conductivity is proportional to the electrons’ diffusivity and is given by the Kubo formula [67]: ∞

σ ij = c0 ∫ e −t /τ vi (t ) v j (0) dt , 0

(5.12)

where c0 = ne 2 / EF , n is the electron density, EF is the Fermi energy, τ is the electron mean scattering time for the system without the AD, i and j subscripts represent the directions x and y. The term vi (t ) v j (0) is the velocity-velocity correlation function averaged over all the particles, where v j (0) is the electron’s initial velocity in direction j. We run the simulation from t = 0 to 10τ, divided into 10,000 discrete time intervals, and perform the numerical integration of Eq. (5.12). Once we obtain the conductivity tensor I I components σ ij , we can calculate the resistivity through the inverse relationship, ρ = σ −1 . The longitudinal MR ρ xx is given by:

ρ xx =

σ yy . σ xxσ yy − σ xyσ yx

(5.13)

The results of our simulations of ρ xx for the X and Y valleys are presented in Fig. 5.6. The MR traces indeed show peaks at or near the expected values, namely orbits X1, X2, X3 for the X-valley and Y1 for the Y-valley. This observation evinces the conjecture outlined in the preceding section that the commensurability peaks are observed only when the orbit diameter matches an integer multiple of the AD lattice period along the direction perpendicular to the current.


41

Figure 5.6 MR obtained from numerical simulations (smooth curves are guides to the eye). Vertical lines indicate the expected positions of the peaks for orbits X1, X2, X3 and Y1. We assumed equal densities for the two valleys and a current along the x-direction. Inset: Schematic of the commensurate orbits. B0 is the magnetic field of the first commensurate orbit if the Fermi contour were circular.

Furthermore, the simulation elucidates the transport processes that give rise to the peaks in the MR traces. At the commensurate conditions, we can classify the electron orbits into three types as shown in Fig. 5.5. The pinned orbits tend to localize the electrons in space, practically removing them from the conduction process, and thus increase the resistance. The chaotic orbits at the commensurate condition too tend to localize the electrons around one or more ADs and can also lead to enhanced resistance. The skipping orbits (or runaway trajectories) clearly increase the conductivity in the transverse direction ( σ yy ) as shown in Figs. 5.5 (a) and (b). Since ρ xx ∝ σ yy , this effect enhances the longitudinal resistivity further and its contribution may be the most significant. This explains our previous conjecture that commensurability peaks are determined by the lattice period along the direction perpendicular to the current. The behavior of the skipping orbits could also explain the difference in the relative

42

5.5 DATA ANALYSIS

strengths of the X1 and Y1 peaks in Fig. 5.6. For orbit Y1, the skipping orbits are harder to occur since the electrons come close to colliding with the ADs in the adjacent column, thus breaking the skipping trajectories. In the case of skipping orbits for X1, on the other hand, the electrons practically skip along in a free space due to their skinny orbits. Since more electrons can follow such trajectories they therefore enhance the conductivity along the y-direction leading to a stronger peak in the longitudinal resistivity.

5.5 DATA ANALYSIS After presenting numerical simulations that provide insight into the transport in our system, we proceed with the analysis of our experimental data. We associate peak A in our data of Figs. 5.3 and 5.4 with the shortest orbit X1 in Fig. 5.2 (c). From the field position of this peak, and if we assume that the electron density of the X valley is half the total density, we can directly obtain a value for the anisotropy (ratio of the major to minor axes diameters)3 of the elliptical orbits in our system, thus obtaining the effective mass ratio ml / mt . However, there is a finite imbalance between the X and Y valley densities in our sample. Such imbalances can occur because of anisotropic strain in the plane of the sample and are very often present in AlAs 2DESs [11,69]. Therefore, we present here an analysis to determine the ml / mt ratio independent of the density imbalance. Consider a primary, commensurate orbit whose diameter in the direction perpendicular to the current is equal to the AD lattice period [orbits X1 and Y1 in Fig. 5.2 (c)]. These would give rise to MR peaks at fields BP = 2=k F / ea where k F is the Fermi wavevector along the current direction. For the X and Y valleys, these wavevectors are k F , X and k F ,Y , respectively, as shown in Fig. 5.2 (b). For an elliptical Fermi contour, they are related to the densities of the X and Y valleys, nX and nY, via the following relations: k F2 , X = 2π n X ml / mt ,

k F2 ,Y = 2π nY mt / ml .

(5.14)

Note that the total density nT = n X + nY and the valley imbalance ∆n = n X − nY . We can obtain nT from the Shubnikov-de Haas oscillations of the MR at high magnetic fields or from a measurement of the Hall coefficient. Now consider orbit X1 as shown in Fig. 5.2 (c). Its associated MR peak position BP , X 1 is given as: BP2,X1 =

3

h2 ml / mt ( nT + ∆n ) . πe2 a 2

The ratio of major to minor axes of the elliptical orbits is given by ml / mt .

(5.15)

5.5 DATA ANALYSIS

43

We use this expression to analyze our data. We assign peak A in our data to orbit X1 and plot the square of its field position BP2, A as a function of the total density nT in Fig. 5.7. It is clear that for all four AD lattice regions, BP2, A varies linearly with nT as expected from Eq. (5.15). Moreover, we obtain the slopes β , and the intercepts of the lines in Fig. 5.7 by performing a least-squares fit of each data set. Note that according to Eq. (5.15), β = h 2 ml / mt / πe 2 a 2 , and the intercept is equal to β ∆n .4 Finally, we plot β as a function of a −2 in the inset of Fig. 5.7. This figure shows that, consistent with the prediction of Eq. (5.15), β indeed depends linearly on a −2 and the line has a zero intercept.

Figure 5.7 Summary of the density dependence of BP2, A for all four AD regions in sample M415B1; BP , A is the position of peak A observed in MR traces. The straight lines are linear fits using Eq. (5.15). The error bars reflect the uncertainty in the peak positions which were determined by subtracting second-order polynomial backgrounds from the MR traces. Inset: Slope (β) of the BP2, A vs. nT lines of the main figure are plotted as a function of a −2 . The dashed-line is a linear fit to the data. 4

Here we assume that ∆n is fixed in the density range of interest.

44

5.5 DATA ANALYSIS

From the slope, ∆β / ∆ ( a −2 ) , of the line in Fig. 5.7 inset, we can deduce the effective mass anisotropy ratio: h4 ml / mt = 2 4 πe

2

 ∆β   ∆( a −2 )  .  

(5.16)

Note that this mass anisotropy ratio is related to the slope of the line in the inset of Fig. 5.7 by a pre-factor containing only physical constants ( h 4 / π 2 e 4 ). Our data analysis and determination of this ratio is therefore insensitive to parameters such density imbalance between the two valleys. From data of Fig. 5.7 we obtain ml / mt = 5.2 ± 0.4 , in very good agreement with the ratio ml / mt = 5.2 ± 0.5 determined from the ballistic transport measurements in AlAs 2DESs subjected to one-dimensional, periodic potential modulations [69] as described in Chapter 4.

Figure 5.8 Density imbalance deduced from the Fourier analysis of the Shubnikov-de Haas oscillations for the 0.8 µm AD region (sample M415B1). Inset: The Fourier spectrum with peaks associated with half total density and half (X and Y) valley densities. The half valley density peaks nX/2 and nY/2 at f = 5.9 and 3.3 T correspond to a valley imbalance of ∆n = 1.3×1011 /cm2.

5.5 DATA ANALYSIS

45

A few other features of the data presented here are noteworthy. From the intercepts of the linear fits in Fig. 5.7 we can determine the valley density imbalance for each AD lattice. Such analysis gives ∆n = 1.5, 1.2, 1.0, and 1.3 ×1011 /cm2 (± 0.2×1011 /cm2) for the AD regions with a = 0.6, 0.8, 1.0 and 1.5 µm, respectively. Such a variation of valley imbalance for different AD regions may come from non-uniform residual strain across the sample. (Fortunately, the values of ∆n or their variations from one region to another do not affect the ml / mt value as determined from our analysis in Fig. 5.7). Note that, because of the close proximity of the different AD lattice regions, we expect this variation to be small, consistent with the ∆n values deduced from the above analysis. We can also deduce the valley imbalance from the Fourier analysis of the Shubnikov-de Haas oscillations measured across the AD regions [24], provided that the valley density peaks in the Fourier spectrum are well developed and well separated. We obtained such data for

Figure 5.9 Summary of the density dependence of BP2 of peak A and B for the AD regions in sample M415B1: (a) a = 0.6 µm and (b) a = 0.8 µm. The lines X1 are linear fits to peak A position using Eq. (5.15). Lines X2, X3 and Y1 are the predicted peak positions calculated using equations similar to Eq. (5.15).

5.6 SUMMARY

46

the a = 0.8 um AD region at high density as shown in Fig. 5.8,5 where we deduce ∆n = 1.3×1011 /cm2, consistent with ∆n = 1.2×1011 /cm2 deduced from the intercept of the linear fit in Fig. 5.7. As for peak B, it is tempting to associate it with orbits X2 or Y2 in Fig. 5.2 (c). This is qualitatively consistent with the data of Fig. 5.3, which indicate that peak B moves to smaller values of magnetic field as the period of the AD lattice is made larger. Moreover, peak B becomes weaker with increasing AD lattice period and disappears for the largest period a = 1.5 µm. This is also consistent with the larger size of the X2 and Y1 orbits (compared to the X1 orbit), and the fact that for a = 1.5 µm, the lengths of these orbits become large compared to the electron mean-free-path. Quantitatively, using the values of ml / mt and ∆n obtained above, we can modify Eq. (5.15) and determine the expected peak positions associated with the X2 and Y1 orbits.6 As illustrated in Fig. 5.9, we find that the predicted peaks for X2, X3 and Y1 orbits are quite close to each other in field7 and approximately straddle the observed positions of peak B in Fig. 5.3. It is possible then that peak B may originate from a superposition of X2, X3 and Y1 peaks that cannot be resolved in our experiment. We cannot rule out, however, that peak B may be strongly influenced by non-linear orbit resonances in the system. Such resonances are known to occur for orbits with long trajectories in the presence of a smooth AD potential [54].

5.6 SUMMARY In summary, we performed ballistic transport experiments in AD lattices imposed on an AlAs 2DES where the electrons occupy two valleys with anisotropic Fermi contours. The low-field MR traces exhibit two sets of peaks. From the analysis of the positions of the peak associated with a commensurate orbit with the shortest trajectory [X1 orbit in Fig. 5.2 (c)], we deduced the effective mass anisotropy ratio ml / mt = 5.2 ± 0.4 , a fundamental parameter of the AlAs conduction-band structure that cannot be directly measured from other transport experiments. This ratio is consistent with the ratio deduced from the measurements of the commensurability oscillations described in the previous chapter. 5

6

7

We have repeated similar measurement for lower densities and for other AD regions, unfortunately this is the only data set where we observe well-developed and well-separated valley density peaks. To calculate peak positions for orbits X2 and X3, simply replace a with 2a and 3a in Eq. (5.15). For orbit Y1, replace ml / mt with mt / ml and ∆n with −∆n . In Fig. 5.9 the expected BP for Y1 orbits are lower than BP for X3 because nY < nX in our experiments.

6CHAPTER QPC in6AlAs 2DES

QUANTUM POINT CONTACT IN ALAS 2DES In this chapter we report experimental results on a quantum point contact (QPC) device fabricated in a wide AlAs quantum well that exhibits quantized conductance. We present a model to describe the quantized energy levels of the system and the role of the in-plane valleys in the transport, and briefly discuss a potential application of such a point contact device as a “valley-filter”.

6.1 QUANTUM POINT CONTACT A QPC is a point-like constriction that connects two electrically conducting regions with typical size (channel length and width) in the nano to micrometer range. In QPCs fabricated in high mobility 2D carrier systems, transport across the constriction is ballistic and the small constriction width also induces a quantization effect which is manifested in the quantized conductance. The quantized conductance in a QPC device is a hallmark of quantum mechanical ballistic transport phenomena in one-dimensional (1D) channels. The conductance is quantized in units of G0 = 2e 2 / h [Fig. 6.1 (b)] where the factor of two accounts for spin degeneracy, e is the electron charge and h is Planck’s constant [34,70].

Figure 6.1 (a) A split-gate QPC device. (b) The original observation of the quantized conductance effect in a QPC [34].

47

6.1 QUANTUM POINT CONTACT

48

This effect has been observed in many two-dimensional (2D) carrier systems with mean free path longer than the channel length subjected to ballistic 1D constrictions. It arises from the quantization of transverse momentum and full transmission of the 1D modes in the constriction reminiscent of the quantum Hall effect, where the absence of backscattering leads to quantized plateaus in the Hall resistance. Such ballistic wires have been implemented in a multitude of material systems to date; such as split-gate semiconductor QPC [as shown in Fig. 6.1 (a)] [34,70] and wires [71], carbon nanotubes [72], and V-grooved [73] or cleaved edge overgrown AlGaAs heterostructures [74]. In 2D carrier systems, quantized conductance in QPC has been observed in systems such as electrons in GaAs [34,70], SiGe [75,76], GaN [77], InSb [78] and holes in GaAs [79,80]. Our QPC device is fabricated via shallow-etching and depositing a top gate that covers the entire sample and controls the density in both the 2D reservoir and the QPC channel. The conductance vs. gate bias trace of this device exhibits several interesting features,

Figure 6.2 (a) Conductance vs. gate voltage in an AlAs QPC (sample M409K5). The arrow indicates the "0.7 structure". (b) The transconductance dG / dVG . The minima, nearly periodic in 2e 2 / h (except for the "0.7 structure"), indicate developing plateaus in the conductance trace.

6.2 QUANTIZED CONDUCTANCE

49

some of which are puzzling. In the next section, we summarize these observations and briefly describe their origin; the rest of the chapter is devoted to the details of the device and a quantitative explanation of conductance vs. gate bias data.

6.2 QUANTIZED CONDUCTANCE Figure 6.2 presents our main finding. It shows the conductance ( G ) trace of the QPC device measured at T = 0.3 K as a function of the applied gate voltage ( VG ) and the corresponding transconductance trace ( dG / dVG ) that accentuates the features in G. Three features of the data are noteworthy. First, we observe developing quantized conductance steps near integer multiples (N) of 2e 2 / h up to N = 5 ; these steps are exhibited more clearly in the transconductance plot of Fig. 6.2 (b).1 Second, the 2e 2 / h conductance plateaus seem to get stronger monotonically at higher N. Third, we observe a particularly strong plateau near 0.7 × 2e 2 / h . Perhaps the most puzzling feature of Fig. 6.2 data is that, even though the system is expected to have a two-fold valley degeneracy, we do not observe quantized steps at integer multiples of 4e 2 / h but instead at multiples of 2e 2 / h . As we discuss in detail in the remainder of the chapter, this happens because the mass anisotropy breaks the valley degeneracy of the QPC subband energy levels. The valley with larger mass in the QPC (lateral) confinement direction has lower energy and therefore dominates the low-lying subband states. Moreover a small but finite residual valley splitting present in the system further helps lift the valley degeneracy. The overall weakness of the observed plateaus in Fig. 6.2 imply that the QPC energy subband spacings are comparable to k BT , where k B is the Boltzmann constant and T is the system temperature. This is not surprising, considering the rather large effective mass of electrons (compared to, e.g., GaAs 2D electrons), which leads to small subband spacings. The fact that the conductance plateaus get stronger monotonically at higher conductance steps [Fig. 6.2 (b)], on the other hand, is somewhat puzzling. This distinct feature is not observed in conventional surface split-gate QPCs [34,70]. We attribute it mainly to two characteristics of our device. First, since our QPC constriction is defined using shallow-etching and gating (in contrast to only gating using split-gate), we expect 1

Small discrepancies between the actual plateau positions and N×2e2/h levels are present due to background resistances of 100 to 800 Ω, which are typical values for resistance of the 2D reservoir regions.

6.2 QUANTIZED CONDUCTANCE

50

the lateral confinement potential to be strong [81,82], resembling a square well potential (Fig. 6.3). In such a potential, the energy subband spacing ∆EN , N +1 increases at higher N, therefore at a given temperature, the subband spacing will be better resolved. Second, the electron mean-free-path in our device is comparable to the length of the channel. As we increase the gate voltage, the density in the QPC channel increases, resulting in higher electron mobility and longer electron mean-free-path [11]. This results in a better transmission across the QPC and thus a better plateau formation at higher VG. We note that, a monotonic increase in strength of 2e 2 / h plateaus suggests that the quantized energy levels originate from a single valley. Any accidental degeneracy between X and Y valley 1D energy levels in the QPC would lead to non-monotonic energy level spacings and would cause a deterioration or missing of conductance plateaus, in contrast to the monotonic increase in plateau strength as observed here.

Figure 6.3 (a) Device schematic of the shallow-etched QPC. The k-space orientation of the in-plane valleys labeled as X and Y is also shown. (b) Confinement potential along the dashed-line in (a). The Fermi energy varies with the gate bias voltage. (c) Device crosssection along the dashed line in (a).

6.3 DEVICE STRUCTURE

51

Finally, the conductance plateau we observe near 0.7 × 2e 2 / h is particularly strong compared to the other plateaus in our QPC device or the "0.7 structure" plateaus reported for QPCs fabricated using other 2D systems. The origin of the "0.7 structure" is still unknown although there is a general consensus that, unlike the integer plateaus, it may be caused by electron-electron interaction [81,83-87]. Our observation is consistent with such interpretation as we indeed do expect interaction to be strong in our system because of the heavy electron effective mass.

6.3 DEVICE STRUCTURE We used a modulation-doped AlAs quantum well (QW) grown by molecular beam epitaxy on a (001) GaAs substrate. The well is 11 nm wide, resides 110 nm below the surface and is flanked by undoped and Si δ-doped layers of Al0.4Ga0.6As [Fig. 6.3 (c)] [11]. The QPC is fabricated on a standard Hall bar device, defined by optical lithography. In most QPC devices, the channel is defined by a pair of surface metal split gates to control the channel's electrical width using an applied voltage bias to the gates [34,70]. In such a device the width of the confinement potential changes considerably while the Fermi level EF in the 2DES reservoir remains constant. In a system with large electron effective mass such as in AlAs 2DES, the energy level spacings are expected to be very small, rendering the observation of quantized conductance challenging. We tried a number of split-gate structures with no success in observing quantized conductance steps. Experiments on high quality AlAs quantum wires fabricated via the cleaved-edgeovergrowth technique, have not yielded clear quantized plateaus either [88]. We therefore took an alternative approach and defined the QPC constriction by shallow-etching to introduce a strong QPC confinement, [81,82] and then covered the entire device by a Ti/Au top gate that controls the 2DES density (and thus EF) in the reservoir and in the QPC constriction [Fig. 6.3 (a)]. As we demonstrate later in this chapter, this device structure allows us to directly probe the energy spacing between the QPC quantized energy levels. One disadvantage is that the background resistance, from the series resistance of the 2DES reservoir regions flanking the point contact, varies with the gate voltage; however, this background correction is found to be rather negligible as shown in Fig. 6.2.

6.4 ANALYSIS AND DISCUSSION

52

Our QPC constriction was defined by electron beam lithography to be 300 nm wide and 500 nm long with the transport direction aligned along [100] as shown in Fig. 6.3 (a) (sample M409K5). Due to carrier depletion near the boundary of the constriction, however, the electrical channel width (w) is expected to be smaller. We performed wetetching using H2SO4:H2O2:H2O (1:8:160) solution to a depth of 65 nm to remove the Si dopant layer, thus completely depleting the electrons under the etched region [Fig. 6.3 (c)]. Using illumination and top/back gate biasing [42], we were able to vary the 2DES density n2D between 2.4 and 4.5×1015 /m2 with a low temperature mobility of ~15 m2/Vs, implying a transport mean-free-path of ~1 µm. The measurements were done in a 3 He cryostat system with 0.3 K base temperature and using standard phase sensitive lockin technique in a four-wire configuration.

6.4 ANALYSIS AND DISCUSSION To gain insight into the transport in the QPC and especially to understand the role of the two in-plane valleys, we present a simple model to describe the QPC energy levels by taking into account the QPC geometry and the electron system parameters. Our goal is to accurately explain the values of VG at which we observe the conductance plateaus. We start with an infinite square well model and a constant QPC channel width. However, as we will show, this description is unsatisfactory for our system, and thus later on we refine the model by treating the channel width as a gate voltage-dependent variable. We also corroborate this analysis with the channel width deduced from our low-field magnetoresistance data.

6.4.1 Quantum point contact model Figures 6.4 (a) and (b) present a model for the potential landscape around the QPC. In our device, as we increase the gate voltage, EF increases and crosses the quantized energy levels in the QPC, leading to quantized conductance steps. The Fermi energy can be obtained from the density n2D in the 2D reservoirs on the two sides of the QPC. This density can be determined from Shubnikov-de Haas oscillations and Hall resistance measurements as will be detailed in the next section (Figs. 6.5 and 6.6). We start with the simple assumption that there are two in-plane valleys X and Y with equal population in the 2D reservoirs participating in transport; X and Y refer to valleys with their major axes along [100] and [010] directions (see Figs. 6.3 and 6.4). Given n2D, we can calculate the


53

Fermi energy EF = n2 D / ρ where ρ = 2m∗ π= 2 is the spin- and valley-degenerate 2D density of states; we used m∗ = ml mt . This EF is plotted as a function of VG in Fig. 6.4 (c). From the conductance trace in Fig. 6.2 we can estimate the gate voltages where EF crosses the 1D subbands in the QPC. Since the plateaus are not sharply defined in our experiment we cannot pinpoint exactly where these crossing points are. However the kinks at every 2e 2 / h in the conductance trace are clear. These kinks occur when EF lies

Figure 6.4 (a) Schematic drawing showing the potential landscape surrounding the QPC and the orientations of the two in-plane valleys X and Y. (b) Potential cross-section along the transport direction [dotted line in (a)]. (c) Dependence of the Fermi energy on the gate voltage (sample M409K5). The circles are the expected level crossings. Right inset: The conductance vs. VG trace together with an idealized conductance step model to determine the level crossings (see text). Left inset: The QPC energy levels assuming a fixed channel width of w = 300 nm. It is apparent that the energy levels do not fit the level crossings.


54

in the middle of an energy gap. Thus the level crossings occur approximately at gate voltages half-way in between successive gate voltages where the kinks appear. Using this approximation we construct an idealized conductance steps trace shown by dashed lines in the inset of Fig. 6.4 (c). The positions where the level crossings occur are marked as circles (labelled with their subband indices N = 1 to 5). Figure 6.4 (b) assumes that the QPC has an elevated bottom potential, with an offset ∆EC , with respect to the 2DES reservoir potential. This is to model the fact that the QPC has a pinch-off voltage VP = 0.3 V (Fig. 6.2) even though the density in the 2D reservoir is non-zero at this voltage. We take ∆EC to be equal to EF at VG = VP . In the QPC channel, the X and Y valleys have a different set of quantized energy levels as shown in Figure 6.4 (b). This happens because, thanks to the different valley orientations, the electrons in the X and Y valleys have different effective masses along the lateral confinement direction. This mass anisotropy breaks the degeneracy of the quantized levels in the channel. Since the confinement potential is strong and quantized energies are small due to heavy electron effective mass, we can estimate the 1D subbands' energy levels using an infinite square well model: EN = N 2 = 2 π 2 / 2m∗y w2 where m∗y is the electron mass along the QPC confinement direction; m∗y is equal to ml (mt) for the Y (X) valley. Thus the ground state energy for valley Y is lower than that of X by a factor of ml / mt = 5.5 . Finally, given the channel width of w =300 nm, we calculated the first few subband levels for both X and Y valleys and plot them in Fig. 6.4 (c). It is apparent that the energy levels do not agree with the expected level crossings. This points to the inadequacy of our simple model and demands further refinement, which we describe in the next section.

6.4.2 Determination of QPC channel density and width Since the constriction is defined by etching, the removal of the dopant layer, together with the Fermi level pinning effect at the exposed surface, induces depletion regions that narrow the QPC channel. Simple theoretical estimates, confirmed by experimental results, indicate that the depletion width decreases with increasing 2DES density [89,90]. In addition, as seen in Fig. 6.3 (c), the top gate covering the shallow-etched region comes to close proximity of the QPC channel (somewhat similar to a split-gate device) so that changing the gate bias may also influence the channel width electrostatically. Therefore, we expect that the channel width w would increase at higher VG where the density is higher.

55


It is possible to estimate the channel width from the QPC magnetoresistance (MR) [91]. Figure 6.5 shows the longitudinal MR (Rxx) traces across the QPC and the transverse MR (Rxy) measured in one of the reservoir regions (see Fig. 6.5 inset). The Rxx traces exhibit strong negative MR at low field. This negative MR arises from the suppression of the constriction resistance by the magnetic field in the ballistic regime. As the magnetic field is increased, the electron backscattering rate is reduced and a larger fraction of the edge states are transmitted through the channel, resulting in a smaller resistance [91]. This behavior persists up to a magnetic field BK at which the classical cyclotron diameter equals the channel width. At BK there is a marked change in the MR slope, appearing as a "kink" in the trace (Fig. 6.5) that can be used to estimate the width of the QPC: w = 2=k F / eBK ,

(6.17)

where kF is the Fermi wavevector perpendicular to the width of the QPC [91].

Figure 6.5 Magnetoresistance traces for the QPC device (sample M409K5). Inset shows the contact geometry for these traces. The kink in the Rxx magnetoresistance trace, (marked by the arrow BK), signals the field where the cyclotron orbit fits into the width of the QPC channel. For VG =0.6 V, a Hall resistance (Rxy) trace is also shown.

56


In our QPC there are two possible in-plane valleys, each with an elliptical cyclotron orbit, that could participate in the transport. For the X and Y valleys, the Fermi wavevectors along the transport direction (perpendicular to the channel width) can be written as [69]: k F2 , X = 2π nQPC , X ml / mt

(6.18)

k F2 ,Y = 2π nQPC ,Y mt / ml

(6.19)

where nQPC,X and nQPC,Y are the 2D valley densities in the QPC for the X and Y valleys, respectively. These electron densities in the QPC channel can be determined from the Fourier spectra of the Shubnikov-de Haas (SdH) oscillations as follows.

Figure 6.6 (a) Fourier transform (FT) spectrum of Rxx representing density components in the 2D reservoir and the QPC (sample M409K5). (b) Fourier transform spectrum of dRxy/dB representing density components in the 2D reservoir alone. (c) Summary of the Fourier spectra peaks vs. VG.

Since Rxx is measured across the QPC (Fig. 6.5), it contains SdH oscillation contributions from both the 2D reservoirs and the QPC region [92]. Ideally, we could obtain density information in the 2D reservoirs from MR measurements in a plain 2D region adjacent to the QPC. Unfortunately, there are no contacts available to measure such a region in our device. Thus, to obtain the density in the 2D reservoirs alone, we deduce the SdH


57

oscillations from dRxy / B that represents the "longitudinal MR" following the empirical resistance rule [93]. It has been shown that the Hall resistance is unaffected by the presence of the QPC constriction [91]. The Fourier spectra of Rxx and dRxy / B are shown in Figs. 6.6 (a) and (b). We observe three common frequency components labeled as A, B and D in Fig. 6.6 (a) [A', B',and D' in Fig. 6.6 (b)] in the Fourier spectra of Rxx and dRxy / B traces. These frequency components are associated with total density (A) and half total density (B) in the 2D reservoir region [24] (We will shortly return to peak D in the next section). We assign the frequency component C, which appears only in Rxx Fourier spectrum [Fig. 6.6 (b)], to the total density in the QPC channel. Our justification for this assignment is two-fold. First, the frequency component C does not appear in the SdH spectrum of dRxy / B that represents SdH oscillations from the 2D reservoir alone. Second, this frequency component extrapolates to zero at VG = 0.3 V [Fig. 6.6 (c)] marking the pinch-off voltage

Figure 6.7 QPC channel electrical width (w), deduced from the kink in the magnetoresistance, for two possible cases where either valley X or Y is occupied; these cases are schematically illustrated on the right. Gray area denotes the error band for w from the MR kink data. The open squares and solid line indicate w determined from the QPC model (see text).


58

VP for the QPC, consistent with the onset of the conductance trace (Fig. 6.2). Therefore, from the peak C positions, we can determine the electron densities in the QPC channel using: nQPC = fC × e / h .

Assuming that only one valley is present in the channel, there are two possible cyclotron orbit orientations and the width w can be calculated for each case based on Eqs. (6.17) (6.19) as shown in Fig. 6.7. Using a scanning electron microscope, we confirmed that the QPC geometrical width is indeed 300 nm, consistent with the width intended in our electron-beam lithography process. This implies that the width deduced for X valley is not realistic since it exceeds 300 nm at high gate voltage. We conclude that it is the Y valley that is occupied in the QPC and therefore dominates the transport across the QPC channel. This gives channel widths ranging from 50 to 130 nm (Fig. 6.7), implying depletion widths of 85 to 125 nm on each side of the channel wall. These are reasonable values for depletion widths in this type of structure [89].

6.4.3 Quantum point contact model - revisited Having established that the QPC width increases with increasing gate voltage, we can now refine our QPC model (Fig. 6.8). As mentioned in Section 6.2, the monotonic increase in plateau strength of the conductance trace with increasing VG suggests that the transport in the QPC arises from a single valley. This situation requires a residual valley splitting that further lifts the valley degeneracy and depopulates one of the valleys. Indeed from the Fourier spectrum of dRxy / B in Fig. 6.6 (b), where the spectrum represents SdH oscillation components arising from the 2D reservoir only, the half valley density peak D’ (at ~2.5 T) and half total density peak B’ (at ~7 T) indicate a valley population imbalance. This imbalance corresponds to a valley splitting of ∆EV = 0.5 meV . In the presence of such a residual valley splitting, the Fermi energy of the majority valley is given as EF = n2 D / ρ + ∆EV / 2 , as plotted in Fig. 6.8 (c). The presence of a residual valley splitting in our sample is not unusual. It normally arises from a built-in residual strain introduced during sample cooldown. Furthermore, this valley splitting could be enhanced in the QPC channel. First, it is possible that additional anisotropic strain maybe introduced during the fabrication process steps such as etching and gate evaporation. Second, the additional confinement and lower electron density in the QPC region could lead to stronger exchange interaction and make the system more


59

easily valley polarized. Thus it is not unlikely that the transport in the QPC channel is valley polarized. Therefore, in the refined model, we assume that there is only one valley that dominates the transport in the QPC, in this case the Y valley as suggested from the previous section and Fig. 6.7. Using an infinite square well model with channel width w as an adjustable parameter, we can calculate the 1D energy subbands to fit the expected level crossings as

Figure 6.8 (a) Potential landscape surrounding the QPC. (b) Potential cross-section along the transport direction. A finite residual valley splitting (∆EV) in the 2D reservoir is shown. In the QPC this valley splitting is enough to completely depopulate the X valley. (c) A revised QPC energy level model with variable channel width (sample M409K5). The model fits the QPC energy levels to all observed level crossings.

6.5 SOURCE DRAIN BIAS SPECTROSCOPY

60

shown in Fig. 6.8 (c). We assume the width w is a monotonic and smooth function of the gate voltage. As we increase EF, the QPC channel gets wider and the quantized energy levels drop. We obtain the gate voltage-dependent width w(VG) as the fitting parameter and plot w as a solid curve in Fig. 6.7 for comparison. We find that the widths deduced from the QPC model fall fairly close, within the error band, to the widths deduced from the negative MR kink. This self-consistency indicates that indeed it is the Y valley that dominates the transport in the QPC, while the X valley in the QPC is depopulated due to a residual valley splitting.

The resulting energy level spacings between successive quantized energy levels are ~0.1 meV, very small compared to typical subband spacing of most GaAs QPC devices (~5 to 20 meV), yet still somewhat larger than the thermal energy of 26 µeV at T = 0.3 K. This is consistent with the fact that we observe only rather weak developing, quantized conductance plateaus. We would like to point out that we can interpret the data of Fig. 6.8 with the X valley instead of Y as the source of the quantized energy levels, and mt as the relevant effective mass along the confinement potential. In order to yield the same energy levels, such an interpretation requires a larger channel width, whose values fall near (slightly above) curve X in Fig. 6.7. This happens because the quantized energy levels EN scale as 1/ m∗w 2 and, to obtain the same energy levels, the channel width has to be larger by ml / mt . However, as mentioned in section 6.4.2, the widths deduced for the case of X valley are not realistic as they exceed the physical channel width of 300 nm at high VG.

6.5 SOURCE DRAIN BIAS SPECTROSCOPY Having provided a basic understanding of the QPC conductance data vs. VG in our system, we proceed to focus on the "0.7 structure". A relevant measurement is the finite source-drain bias spectroscopy [77,81,84]. We measured the differential conductance G = dI dVSD by applying a small AC excitation signal superposed on a DC-drain bias (VSD). Figure 6.9 (a) shows that the measured source-drain bias conductance traces exhibit non-linear transport behavior through the QPC. We observe a bunching of the traces along the VSD = 0 line that indicate the formation of plateaus at integer multiples of 2e2/h.


61

These features in the conductance traces can be seen more clearly in the transconductance spectrum dG dVG obtained numerically. This spectrum is displayed in Fig. 6.9 (b) where the blue color indicates minima in dG dVG that correspond to developing conductance plateaus at integer multiples of 2e2/h. If the plateaus are well developed one can extract the energy subband spacings in the QPC [81,94]. Unfortunately, since the plateaus are weak in our device, we cannot deduce the energy subband level spacings from this data. However, one particularly interesting feature of the non-linear conductance measurements, relevant to the "0.7 structure", is the conductance peak around zero VSD, known as the zero bias anomaly (ZBA) [84]. This feature can be observed more clearly in the

Figure 6.9 (a) Differential conductance G = dI/dVSD traces measured at T = 0.3 K (sample M409K5). Each trace corresponds to incremental step of 2 mV in VG starting from 0.3 V at the bottom. (b) The transconductance (dG/dVG) spectrum. The bluish (dark) regions correspond to developing plateaus. The numbers represent the plateaus indices.


62

expanded plot of the VSD bias spectrum as shown in the inset of Fig. 6.10. The ZBA below the first 2e2/h plateau has been observed previously in other systems such as in GaAs [84] and GaN [77] QPCs and has been associated with a Kondo-like correlated state. The width of the ZBA was found to show a dip near the "0.7 structure" and increase monotonically with increasing gate voltage [77,84]. Here, we observe qualitatively similar behavior (Fig. 6.10). It was argued in Ref. [84] that the widths of the ZBA peaks, after the dip, are close to 2k BTK / e where TK is the Kondo temperature of the system. However, we remark on a difference here. It appears in our data that the ZBA persists above the first quantized plateau (2e2/h) and its width keeps increasing, while in the GaAs system the ZBA width diverges as the conductance traces collapse to the first plateau [84]. This could simply be related to the fact that our QPC does not show a strong plateau at 2e2/h.

Figure 6.10 Full-width at half-maximum (FWHM) of the ZBA peak as a function of gate voltage (sample M409K5). The color data points (plus signs) are associated with the conductance trace of the same color in the inset. For reference the conductance trace G vs. VG is replotted. Inset: Differential conductance G = dI/dVSD at low conductance where the ZBA occurs.

6.6 SUMMARY

63

6.6 SUMMARY We studied transport in a QPC device fabricated in a two-dimensional electron system confined to an 11 nm-wide AlAs QW. The QPC is defined by shallow-etched regions and entirely covered by a top gate that controls the density in both the 2D reservoirs and the QPC channel. The conductance trace obtained shows developing quantized conductance plateaus at integer steps of 2e 2 / h . From the density measurement in the 2D reservoirs and in the QPC channel we construct a simple model with a gate-voltage dependent channel width to describe the crossings of the Fermi energy and the QPC quantized energy levels. The QPC model and the channel width determination using the negative magnetoresistance kink corroboratively indicate that the transport in the QPC channel is dominated by the Y valley, the valley with larger mass along the QPC lateral confinement direction. This is a plausible outcome since the lowest QPC subband energy levels should be dominated by the carriers with larger effective mass along the confinement direction. Our results highlight the intricacies of understanding quantized conductance in a system with multiple and anisotropic valleys such as ours. Additionally, we observe a welldeveloped "0.7 structure" that may reflect strong electron-electron interaction in this system thanks to its heavier electron mass. Our study suggests that the anisotropy of the effective mass can be exploited to realize a simple "valley-filter" using QPC device. Here the QPC confinement breaks the valley degeneracy and if we set the conductance at the lowest quantized level we will obtain a valley-polarized transmission. Such a filtering device could be important for the detection or generation of valley-polarized currents and may play an important role in "valleytronics" or valley-based electronics.

C 7 7 HAPTER ANOMALOUS GPR IN ALAS 2DES

ANOMALOUS GIANT PIEZORESISTANCE IN ALAS 2DES WITH ANTIDOT LATTICE In this chapter we present our finding of a novel phenomenon, namely, an anomalous giant piezoresistance (GPR) in AlAs two-dimensional electron system (2DES) with an antidot (AD) lattice. This phenomenon is anomalous in that it cannot be explained in terms of the conventional piezoresistance based on the change in effective mass following the inter-valley electron transfer: it has a stronger piezoresistivity and with an opposite sign. To elucidate the physics behind this effect we perform various magnetoresistance measurements, and construct certain hypotheses. We also do numerical simulations to test our hypotheses and propose a simple model that leads to a fascinating implication: the operating principle of our GPR mimics that of the giant magnetoresistance (GMR) effect which is well known in spintronics. Furthermore, we perform additional device characterizations to investigate the promising potential of our GPR device to be utilized as an ultra sensitive strain sensor.

7.1 INTRODUCTION The piezoresistance (PR) effect, the change of a material’s resistance with an applied mechanical stress, was first discovered in 1856 by Lord Kelvin in metallic materials [95]. In semiconductors, the PR effect has been an important element of research on multivalley materials such as Si and Ge. In fact, the observation of a large PR effect in Si and Ge in 1950s served as early experimental evidence for the existence of multiple, anisotropic valleys in the conduction band [14,96]. The large PR effect in these semiconductors compared to metallic compounds could only be explained by the transfer of electrons between anisotropic conduction band valleys that have different effective masses along the transport direction. The PR effect in semiconductors has been exploited in variety of applications such as very sensitive solid-state strain, pressure and fluid flow sensors [97].

64

65

7.1. INTRODUCTION

There are two main sources for a material’s piezoresistivity: the geometric factor and the material or electronic property factor. The geometric factor accounts for the change in the material’s length and cross sectional area that alters the resistance due to the applied stress. This is the primary source of the PR effect in metallic films. The electronic property factor accounts for the change in the electronic band structure of the material due to the applied stress. In multivalley semiconductors such as Si, Ge and AlAs, the applied stress produces appreciable changes in resistance resulting from the charge transfer between anisotropic conduction band valleys. The strength of piezoresistivity can be measured in terms of a gauge factor (GF) defined as the fractional change in resistance per unit strain:

GF =

dR dρ = 1 + 2ν + , ρ d εx R d εx

(7.1)

Figure 7.1 An idealized (conventional) PR effect due to strain-induced intervalley electron transfer in AlAs 2DES, a simple two-valley system: (a) The PR trace vs. strain calculated using Eq. (7.4). εd is the depopulation strain, beyond which the valley is fully polarized. Note that Rxx ,max / Rxx ,min = ml / mt . (b) The valley transfer model. (c) The valley energy subband model with a valley splitting ∆EV that can be controlled by strain.

66

7.1. INTRODUCTION

where ρ is the resistivity, ν is the solid's Poisson ratio and εx is the longitudinal strain. The 1 + 2ν term represents the geometric factor and d ρ / ρ d εx represents the electronic property factor that becomes dominant in multivalley semiconductors. A typical metal film has GF ~ 2, while semiconductors have GF of –125 to 175 (at room temperature) depending on the type and concentration of the impurities (Ref. [98], p. 147). Large GF as high as ~1500 has been achieved at room-temperature in Si-MOSFETs [99].1 For introduction it is worthwhile to briefly review the (conventional) PR effect in AlAs 2DESs. Recently a large PR effect in (blank or unpatterned) AlAs 2DES was demonstrated with GF as high as 12,000 at low temperature (T = 0.3 K) [16].2 This conventional PR effect can be described in terms of a simple strain-induced charge transfer between two anisotropic valleys [16,23]. For example, assuming the transport direction is along [100], an applied tensile strain along [100] induces electron transfer to the Y-valley [Fig. 7.1 (b)] whose effective mass is smaller along the transport direction, leading to a smaller resistance as shown in Fig. 7.1 (a). This behavior can be described using a simple transport model with two valley channels X and Y, where the conductivity along [100] (at T = 0 K) is given as:  n X τ X nYτ Y  + , m m t   l

σ = e2 

(7.2)

where τx and τy are the electron scattering lifetime, nx and ny are electron densities in the the X and Y valley respectively.3 These valley densities can be controlled by an applied symmetry-breaking strain ε :4,5 ∗

nT m∗ E2 nX = − ε 2 4π = 2

∗

and

nT m∗ E2 nY = + ε, 2 4π = 2

(7.3)

with nT is the total electron density, m* is the electron density-of-states mass, and E2* is 1

2

3 4 5

The large GF in semiconductors, however, comes with drawbacks. In contrast to metallic films, the piezoresistivity of semiconductors is typically very non-linear and sensitive to changes in temperature. In Ref. [16] a maximum GF of 11,700 was reported (for a blank region) and in our sample a maximum GF of 15,000 is found for the blank region. Because of their large magnitude (GF 1000) this PR is called GPR. However, this model ignores inter-valley scattering. Here we define the (symmetry-breaking) strain as ε = ε[100] − ε[010] . These equations can be derived, e.g. for the case of X-valley, from: n X = ( nT − ∆n ) / 2 , where ∗ 2 ∆n = nY − n X = g 2 D ∆EV is the density imbalance, with g 2 D = m / 2π= as the spin-degenerate 2D ∗ density of states and ∆EV = E2 ε as the strain-induced valley-splitting energy.

67

7.1. INTRODUCTION

the “renormalized” AlAs conduction band deformation potential constant.6 The “band” value of E2 determined from optical measurements is (5.8±0.1) eV [26]. To describe the PR curve for the blank region in Fig. 7.1, we can simplify Eqs. (7.2) and (7.3) by assuming isotropic scattering times τ = τ X = τ Y , to: 1 − ε/εd 1 + ε/εd  + ,  ml / m0 mt / m0 

σ =σ0 

(7.4)

valid for −ε d < ε < εd , where σ 0 = e 2τ nΤ / 2m0 and εd = 2π= 2 nT / m∗ E2∗ is the valley depopulation strain. This model provides a reasonable fit to the PR data of the blank region as we will discuss later. In this chapter we present our discovery of an anomalous and enhanced GPR effect in AlAs 2DESs with AD lattices where the PR is stronger and has opposite sign compared to that of the blank region. The experimental setup is summarized in Fig. 7.2. In the following sections, we describe the device fabrication technique, various experimental results with some analysis, as well as a proposed model that attempts to explain such an anomalous GPR effect.

Figure 7.2 The setup of the GPR experiment in the AlAs 2DES with AD lattices: (a) The device is glued to a piezo actuator to apply a tunable strain. Thick arrows indicate the direction of strain with positive piezo bias. (b) The Hall bar consisting of a blank region and three AD-patterned regions with AD lattice periods a = 1, 0.8, and 0.6 µm. (c) The corresponding electron transfer to the Y-valley upon the application of positive piezo bias. 6

The product m*E2* actually depends on the total density nT. It is enhanced compared to its “band value” and increases at lower electron density due to increasing electron-electron interaction. This was recently concluded from our “valley susceptibility” measurements performed on the blank region of the sample presented in this chapter (M409K8). The valley susceptibility χV∗ , is proportional to m*E2*. See Ref. [4] for details.

7.2. DEVICE FABRICATION

68

7.2 DEVICE FABRICATION We performed experiments on a 2DES confined to an 11 nm-wide AlAs quantum well (sample M409K8) whose structure is shown in Fig. 2.3 (b) in Chapter 2. We patterned the Hall bar along the [100] direction using standard photolithography techniques. The device has four Hall bar regions as shown in Fig. 7.2 (b), consisting of three AD lattice regions and a blank region that serves as a control region for the experiments. With electron beam lithography and dry etching, using an electron cyclotron resonance etcher (see section 5.2), we patterned the three AD regions with lattice periods a = 1, 0.8 and 0.6 µm; these are labelled as 1 µm-AD, 0.8 µm-AD, and 0.6 µm-AD respectively as shown in the upper inset of Fig. 7.3. Each AD lattice is an array of holes (ADs) etched to a depth of 300 nm into the sample thus depleting the 2DES in the hole area [the 2DES is at a depth of 100 nm from the top surface as indicated in Fig. 2.3 (b)]. The aspect ratio d / a of each AD cell is ~1:3, where d is the AD diameter.7 We also deposited Ti/Au back and front gates to control the 2DES density in the sample. With the help of brief illumination at low temperatures and gating we can achieve a density range of nT = 2.5 to 7.5×1011 /cm2. By gluing the sample on a piezo-actuator as shown in Fig. 7.2 (a) we can apply the symmetry-breaking strain ε to control the valley population. The transport direction of the sample is aligned with the polling direction of the piezo along [100]. When a positive voltage bias VP is applied to the piezo stack, it expands along [100] and shrinks in the [010] direction and thus induces electron transfer from the X valley to the Y valley as shown in Fig. 7.2 (c). To monitor the strain accurately we used a metal foil strain gauge glued to the back of the piezo. We obtain a value of ε of (4.9±0.2)×10-7 per 1 V of piezo bias VP. This quantity is referred to as the piezo strain factor ∆ε/∆V whose measurement is described in Appendix D. The transport measurements are performed in a standard pumped 3He cryostat system with a base temperature of T = 0.3 K.

7.3 EXPERIMENTAL RESULTS AND DISCUSSION In this section we present our main experimental findings: the GPR effect in the AD and blank regions, the density information in both the AD and blank regions derived from high-field magnetoresistance, and finally the AD commensurability peaks found in the low-field magnetoresistance. 7

The shape of each AD is actually square by design, however, due to imperfection in the electron beam lithography and the dry etching process, the ADs are effectively circular (see the upper inset of Fig. 7.3).

7.3. EXPERIMENTAL RESULTS AND DISCUSSION

69

Figure 7.3 The GPR effect in an AlAs 2DES in both the blank region and the AD regions (sample M409K8).8 The dotted line is a fit of Eq. (7.4) to the blank region’s PR. From separate measurements in the blank region we determined the valley-balanced point (VP0 = − 250 V, thus designated as ε = 0 ) and the valley-polarized point (VP,d = 50 V), beyond which only the Y-valley is occupied.9 Note that this condition does not necessarily apply to the AD regions which might be under different residual strain due to the presence of the AD lattice. 8

9

The strain (bottom axis) is calculated from the piezo bias (top axis) using: ε = (VP − VP 0 ) × ∆ε/∆VP , thanks to the linear relationship between strain and piezo bias at low temperature. Here VP 0 = −250 V is the valley-balanced point, and ∆ε/∆VP =(4.9±0.2)×10-7 /V is the piezo strain factor. As detailed in Ref. [4], the valley-balanced point was determined using the “valley-coincidence” technique, where we essentially track the oscillations in the PR at a given filling factor (at a fixed B⊥ ) originating from the Landau level crossings of the two valleys.


70

7.3.1 The piezoresistance of the AD and blank regions Figure 7.3 presents our main finding. The PR in the blank region shows the anticipated behavior, i.e., the resistance drops as the electrons are transferred to the Y-valley whose mobility is higher along the current direction. Furthermore, beyond the valley depopulation point (VP > 50 V for the sample and density of Fig. 7.3) the resistance starts to saturate at a low value as the intervalley electron transfer ceases. This is the conventional PR effect in AlAs 2DESs as described in Section 7.1. The dotted line in the Fig. 7.3 shows the best fit of the data to Eq. (7.4) and demonstrates a reasonable agreement between the experimental data and our simple model.10

Figure 7.4 The GF vs. strain that characterizes the strength of the piezoresistivity calculated from the data of Fig. 7.3.11 The higher GF of the AD regions demonstrates the enhanced GPR effect. 10

11

−4

Given ε d = 1.48 × 10 determined from the “valley-coincidence” measurements [4], the curve fit yields −4 σ 0 = 4 × 10 S . Apparently the resistance still changes beyond ε d , this behavior could be attributed to finite temperature effect that “smears out” the PR trace (see, e.g., Ref. [23], Sec. 5.3). The GF is calculated from our data using: GF = ( dRxx / d εx ) / Rxx = (1 + ν )( dRxx / d ε ) / Rxx , with ν = 0.32 as the Poisson ratio of AlAs. We use the relationship: ε = (1 + ν ) εx .


71

But the AD regions show an enhanced GPR effect with opposite sign (positive) compared to the blank region! The largest effect is found in the 1 µm-AD region where there is as much as 10× change in resistance in the range of interest (in contrast, the blank region only shows a 2.2× change). The larger change corresponds to a higher GF as shown in Fig. 7.4, with a maximum value of 20,000. The fact that there is a larger change in resistance and with an opposite sign compared to that of the blank region suggests that this GPR effect cannot be accounted by a simple change in effective mass alone as in the case of the conventional PR in the blank region. Note that the strength of the PR effect is reduced at smaller lattice periods, e.g. for the 0.6 µm-AD region, GF ,max is only ~10,000. And finally, for all AD regions, the

Figure 7.5 Finite element simulation of the strain distribution in a 2D medium perforated with an AD lattice. The boundary condition is such that the applied tensile stresses at the left and right sides give a unit of strain at x, y → ±∞ . The boxes A and B highlight the areas of enhanced strain.


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piezoresistive behavior persists beyond the valley depopulation point (VP > 50 V) of the blank region; in contrast the PR of the blank nearly saturates for VP > 50 V. The above observations highlight the remarkable difference between the PR effect in the blank and the AD regions and present an interesting puzzle. Clearly the presence of the AD lattice is responsible for this behavior. We suspect that the AD lattice significantly modifies the strain distribution in the AlAs 2DES. To understand the strain distribution in the AD regions we performed a simple finite element method (FEM) simulation of a plane strain problem for a 2D medium perforated with an array of holes.12 We assume a unit tensile stress σ x = σ 0 , in both left and right faces that produces one unit of strain ( ε0 ) at x, y → ±∞ .13 In other words, if there were no AD lattice, the strain would be uniform everywhere with a unit magnitude ( ε = ε0 ). The simulation is done using FEMLAB which runs in the MATLAB environment. The result of this FEM simulation is shown in Fig. 7.5. First, we observe that away from the AD lattice ( x, y → ±∞ ), the strain ε tends to reduce to one unit (cyan color) as expected. Note that positive (negative) strain means that electrons preferentially occupy the Y-valley (X-valley). We see that in the AD lattice region there is a non-uniform strain distribution due to the presence of the AD lattice. Furthermore there are regions of enhanced strain in the AD region as shown in boxes A and B in Fig. 7.5. For example in the upper and lower periphery of the AD (box A) there is a very high concentration of positive strain, as much as 3×, compared to the background (unit) strain. This effect is a manifestation of St. Venant’s principle,14 a well-known principle in the field of mechanics of materials. The principle states that, the distribution of stress and strain in a body is altered only near the regions where a load application or a disturbance occurs, the AD lattice in our case. We will return to this strain distribution picture by incorporating the results of more experimental data as described in the following sections. We add that our simulation of Fig. 7.5 is for a 2D system, however we expect that in a system which contains an AD lattice at its tops surface like our sample, the strain profile is qualitatively similar to what is shown in Fig. 7.5.

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13 14

There is an analytical solution for a 2D plane strain problem with a single hole. See, e.g., Ref. [100], p. 184. We assume very small strain (